Harmonic conformable refinements of Hermite-Hadamard Mercer inequalities by support line and related applications

ABSTRACT

We establish new conformable fractional Hermite-Hadamard (H–H) Mercer type inequalities for harmonically convex functions using the concept of support line. We introduce two new conformable fractional auxiliary equalities in the Mercer sense and apply them to differentiable functions with harmonic convexity. We also use Power-mean, Hölder’s and improved Hölder inequality to derive new Mercer type inequalities via conformable fractional integrals. The accuracy and superiority of the offered technique are clearly depicted through impactful visual illustrations. We also use our technique to derive new estimates for hypergeometric functions and special means of real numbers that are more precise than existing ones. Some applications are provided as well. Our results generalize and extend some existing ones in the literature.

KEYWORDS:

• Harmonic Convex functions
• Jensen Mercer Inequalities
• Fractional Calculus
• Improved Holder type inequalities
• Quadrature Estimations

1 Introduction

Convexity holds significance, in both pure and applied mathematics as it allows for the generalization of existing results. For over 60 years researchers have extensively studied convex functions, which have applications across fields like mathematics, physics, economics and optimization. Research on functions and integral inequalities remains an area of interest. The study of convex functions and integral inequalities remains an area of research leading to significant advancements in convex analysis and fractional integral inequalities. Harmonic convex functions represent a development within the realm of convex functions with diverse applications in mathematics and physics. For instance, they find utility in theory, electrostatics and fluid mechanics. Numerous efforts have been dedicated to extending the concept of convexity and establishing (H–H) type inequalities, for them. A variety of unique and noteworthy inequalities can be obtained from harmonic convex functions. In the paper (ÍşCan 2014; Dragomir 2017; Latif et al. 2017; Gao et al. 2020; Butt et al. 2021) one can explore many excellent inequalities pertaining to harmonically convex functions. The classical definition of a convex mapping is as follows:

Definition 1.1.

Roberts and Varberg (1973) A function $\mathit{\varphi }:\mathbb{I}\subseteq \mathbb{R}\to \mathbb{R}$ is called convex, if

(1.1) $\mathit{\varphi }(\mathit{\gamma }{\omega }_1+(1-\mathit{\gamma }){\omega }_2)\le \mathit{\gamma \varphi }({\omega }_1)+(1-\mathit{\gamma })\mathit{\varphi }({\omega }_2).$(1.1)

For each ${\omega }_1,{\omega }_2\in \mathbb{I}$ and $\mathit{\gamma }\in [0,1]$ holds.

The following convex mapping qualities are employed in the main outcomes.

Definition 1.2.

Roberts and Varberg (1973) A mapping $\mathit{\varphi }$ on interval $\mathbb{I}$ has a support at point $k_0\in \mathbb{I}$, if there exists an affine function $\mathit{A}(\mathit{k})=\mathit{\varphi }(k_0)+\mathit{c}(\mathit{k}-k_0)$ such that $\mathit{A}(\mathit{k})\le \mathit{\varphi }(\mathit{k})$ for all $\mathit{k}\in \mathbb{I}$. The graph of support function A is support line for function $\mathit{\varphi }$ at point $k_0$.

Theorem 1.1.

Roberts and Varberg (1973) $\mathit{\varphi }:({\omega }_1,{\omega }_2)\to \mathbb{R}$ is a convex function iff there is at least one line of support for $\mathit{\varphi }$ at each ${\kappa }_0\in ({\omega }_1,{\omega }_2)$.

Definition 1.3.

A mapping $\mathit{\varphi }:\mathbb{I}\subset \mathbb{R}\mathrm{\setminus }\{0\}\to \mathbb{R}$ is harmonically convex $(\mathit{\varphi }\in \mathit{HC}(\mathbb{I}))$, if

(1.2) $\mathit{\varphi }(\frac{{\omega }_1{\omega }_2}{\mathit{\gamma }{\omega }_1+(1-\mathit{\gamma }){\omega }_2})\le \mathit{\gamma \varphi }({\omega }_2)+(1-\mathit{\gamma })\mathit{\varphi }({\omega }_1),$(1.2)

for all ${\omega }_1,{\omega }_2\in \mathbb{I}$ and $\mathit{\gamma }\in [0,1]$.

Remark 1.1. (Dragomir 2017) Let $[{\omega }_1,{\omega }_2]\subset \mathbb{I}\subseteq (0,\mathrm{\infty })$, if function $\mathfrak{h}:[\frac{1}{{\omega }_2},\frac{1}{{\omega }_1}]\to \mathbb{R}$ defined as $\mathfrak{g}(\mathit{\kappa })=\mathit{\varphi }(\frac{1}{\kappa })$, then $\mathit{\varphi }$ is harmonically convex on $[{\omega }_1,{\omega }_2]$ iff $\mathfrak{g}$ is convex on $[\frac{1}{{\omega }_2},\frac{1}{{\omega }_1}]$.

Proposition 1.1.

ÍşCan (2014) If $\mathit{\varphi }$ : (0,α)$\to \mathbb{R}$ is convex and nondecreasing on (0,α), then $\mathit{\varphi }\in \mathit{HC}(\mathbb{I})$

The classical (H–H) inequality, a well-known inequality in convex function theory, has a geometric meaning as well as various practical applications. It asserts that if $\mathit{\varphi }$ is a convex mapping defined on the interval $[{\omega }_1,{\omega }_2]$, then:

$\mathit{\varphi }\left(\frac{{\omega }_1+{\omega }_2}{2}\right)\le \frac{1}{{\omega }_2-{\omega }_1}{\int }_{{\omega }_1}^{{\omega }_2}\mathit{\varphi }(\mathit{\kappa })\mathit{d\kappa }\le \frac{\varphi ({\omega }_1)+\mathit{\varphi }({\omega }_2)}{2}.$

The inequality holds in reversed direction if the function is concave on $[{\omega }_1,{\omega }_2]$. Several variants have been developed regarding Mercer (H–H) type inequalities (see (Bohner et al. 2023; Budak and Kara 2023; Sitthiwirattham et al. 2024)). (H-H) inequality is the utmost important and extensively used result involving harmonic convex functions. In (ÍşCan 2014), I$\mathrm{s}$can gave (H–H) type inequalities for harmonically convex mapping stated as

Theorem 1.2.

Let $\mathit{\varphi }\in \mathit{HC}(\mathbb{I})$ and ${\omega }_1,{\omega }_2\in \mathbb{I}$ with ${\omega }_1<{\omega }_2$. If $\mathit{\varphi }\in \mathit{L}[{\omega }_1,{\omega }_2]$, then

(1.3) $\mathit{\varphi }\left(\frac{2{\omega }_1{\omega }_2}{{\omega }_1+{\omega }_2}\right)\le \frac{{\omega }_1{\omega }_2}{{\omega }_2-{\omega }_1}{\int }_{{\omega }_1}^{{\omega }_2}\frac{\mathit{\varphi }(\mathit{\kappa })}{{\kappa }^2}\mathit{d\kappa }\le \frac{\varphi ({\omega }_1)+\mathit{\varphi }({\omega }_2)}{2}.$(1.3)

The Jensen inequality is a generalization of the notion of convexity for n-convex combinations. Dragomir in (Dragomir 2015), gave the notable Jensen type inequality for harmonic convex functions in such a way that, for any $\mathit{\varphi }\in \mathit{HC}(\mathbb{I})$ and any non-negative weights ${\gamma }_j\in [0,1]$, $(\mathit{j}=1,2,3,\dots .,\mathit{k})$ with ${\sum }_{j=1}^k{\gamma }_j=1$, we have:

(1.4) $\mathit{\varphi }(\frac{1}{{\sum }_{j=1}^k\frac{{\gamma }_j}{{\mathrm{\ell }}_j}})\le \sum _{\mathit{j}=1}^k{\gamma }_j\mathit{\varphi }({\mathrm{\ell }}_j).$(1.4)

Jensen’s inequality is widely recognized as a fundamental component within the realm of mathematical inequalities. The concept has extensive application in the fields of mathematics, statistics, and information theory, enabling the derivation of several significant inequalities, including the arithmetic-geometric mean inequality, Minkowski inequality, Höder inequality, and Ky Fan’s inequality. In 2020, Baloch et al. presented a new variant of Jensen’s inequality. This variant is defined as

Theorem 1.3.

Let $\mathbb{I}=[{\omega }_1,{\omega }_2]\subseteq (0,\mathrm{\infty })$ be an interval. If $\mathit{\varphi }\in \mathit{HC}(\mathbb{I})$, then

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-{\sum }_{j=1}^k\frac{{\gamma }_j}{{\mathrm{\ell }}_j}})\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\sum _{\mathit{j}=1}^k{\gamma }_j\mathit{\varphi }({\mathrm{\ell }}_j),$

for all ${\mathrm{\ell }}_j\in [{\omega }_1,{\omega }_2]$ and ${\gamma }_j\in [0,1],(\mathit{j}=1,2,3,\dots .,\mathit{k})$ with ${\sum }_{j=1}^k{\gamma }_j=1$.

The field of fractional calculus gained significant attention from scholars across various academic areas due to its intriguing nature. This field of research explores the concepts of differentiation and integration of arbitrary order within the realm of mathematics. Academic researchers are attracted to this particular topic due to its ability to provide meaningful outcomes through the utilization of fractional operators for the purpose of modelling real-world problems. Fractional integral and derivative operators have developed over time (Khalil et al. 2014; Atangana and Baleanu 2016). Thus, these operators have been considered as one of the most powerful tools in the area of mathematical modelling. Many engineering, physical, chemical, and biological phenomena can be modelled by employing differential equations containing fractional derivatives (Atangana 2017; Gomez-Aguilar and Atangana 2022). Prominent academics like as R.P. Agrwal and D. Baleanu have made significant contributions to the enormous body of research in this particular topic. The publication entitled ‘Fractional calculus in the sky’ provides a comprehensive survey of the progress made in this particular field (Baleanu and Agarwal 2021). Furthermore, the field of inequalities has been enhanced by the discovery of several variants generated from distinct convexities, as demonstrated by their applications (Sarikaya et al. 2013; Chen and Katugampola 2013; Set et al. 2021; Sun 2021; Butt et al. 2021; Yuan et al. 2023). Over the course of several decades, there has been a notable emphasis on the investigation of diverse approaches aimed at establishing integral inequalities associated with harmonic convexity within the field of fractional calculus. In their publication (IşCan and Wu 2014), $\mathit{I}$$\mathrm{s}$can and S. Wu established a set of inequalities of the (H–H) type for a function $\mathit{\varphi }\in \mathit{HC}(\mathbb{I})$. These inequalities are expressed in the form of integrals, as demonstrated below.

Theorem 1.4.

Let $\mathit{\varphi }:\mathbb{I}=[{\omega }_1,{\omega }_2]\subseteq (0,\mathrm{\infty })\to \mathbb{R}$ be a function such that $\mathit{\varphi }\in \mathit{L}[{\omega }_1,{\omega }_2]$ with $0<{\omega }_1<{\omega }_2$. If $\mathit{\varphi }\in \mathit{HC}(\mathbb{I})$ on the interval $\mathbb{I}=[{\omega }_1,{\omega }_2]$, then

$\mathit{\varphi }\left(\frac{2{\omega }_1{\omega }_2}{{\omega }_1+{\omega }_2}\right)\le \frac{\mathrm{\Gamma }(\mathit{\vartheta }+1)}{2}{\left(\frac{{\omega }_1{\omega }_2}{{\omega }_2-{\omega }_1}\right)}^{\mathit{\vartheta }}\left[{\mathcal{J}}_{\frac{1}{{\omega }_1}-}^{\mathit{\vartheta }}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\omega }_2})+{\mathcal{J}}_{\frac{1}{{\omega }_2}+}^{\mathit{\vartheta }}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\omega }_1})\right]$

$\le \frac{\varphi ({\omega }_1)+\mathit{\varphi }({\omega }_2)}{2}.$

For further recent developments one can (see (Rashid et al. 2020; Latif et al. 2022; Butt et al. 2022)). We need some important special functions used for our study.

Definition 1.4. (Tunc et al. 2015)

1. The Beta function has real number domain and defined as

(1.5) $\mathit{\beta }({\omega }_1,{\omega }_2)=\frac{\mathrm{\Gamma }({\omega }_1)\mathrm{\Gamma }({\omega }_2)}{\mathrm{\Gamma }({\omega }_1+{\omega }_2)}={\int }_0^1{\gamma }^{{\omega }_1-1}(1-\mathit{\gamma }{)}^{{\omega }_2-1}\mathit{d\gamma },{\omega }_1,{\omega }_2>0.$(1.5)

2. The Hypergeometric function is defined as:

(1.6) ${ }_2\mathfrak{F}1({\omega }_1,{\omega }_2;\mathfrak{c};\mathfrak{z})=\frac{1}{\beta ({\omega }_2,\mathfrak{c}-{\omega }_2)}{\int }_0^1{\gamma }^{{\omega }_2-1}{\left(1-\mathit{\gamma }\right)}^{\mathfrak{c}-{\omega }_2-1}(1-\mathfrak{z}\mathit{\gamma }{)}^{-{\omega }_1}\mathit{d\gamma },$(1.6)

where $\mathfrak{c}>{\omega }_2>0,|\mathfrak{z}|<1$.

Lemma 1.1.

Prudnikov et al., (1998) For $0<\mathit{\xi }\le 1$ and $0<{\omega }_1\le {\omega }_2$ we have

$\left|{{\omega }_1}^{\mathit{\xi }}-{{\omega }_2}^{\mathit{\xi }}\right|\le ({\omega }_2-{\omega }_1{)}^{\xi }.$

Conformable fractional operators are a new class of fractional operators that have been recently proposed to overcome some of the limitations and complications of the traditional fractional operators, such as the Riemann-Liouville and Caputo derivatives. Conformable fractional operators are based on the basic limit definition of the derivative and preserve some of the properties of the integer-order calculus, such as the product rule, the chain rule, and the Leibniz rule. Conformable fractional operators have been used to develop new results and applications in various fields of mathematics and science, such as differential equations, stability analysis, nonlinear observers, and approximation methods.

In their research (Nisar et al. 2019), Nisar et al. extended Minkowski and (H–H) inequalities using conformable fractional integral operator. The results are generalization of the integral inequalities obtained by Dahmani and Bougoffa cited in the literature. In their research (Adil Khan et al. 2018), Khan et al. presented an advanced version of the renowned (H–H) inequality, utilizing newly defined Generalized Conformable fractional operators. In (Abdeljawad 2015), left and right Conformable fractional integrals is defined as

Definition 1.5.

For $\mathit{\xi }\in (\mathrm{\aleph },\mathrm{\aleph }+1],\mathrm{\aleph }=0,1,2,\dots .,\mathit{\beta }=\mathit{\xi }-\mathrm{\aleph },{\omega }_1,{\omega }_2\in \mathbb{R}$ with ${\omega }_1<{\omega }_2$ and $\mathit{\varphi }\in \mathit{L}[{\omega }_1,{\omega }_2]$. The left and right Conformable fractional integrals ${\mathcal{I}}_{\xi }^{{\omega }_1}$ and ${ }^{{\omega }_2}{\mathcal{I}}_{\xi }$ of order $\mathit{\xi }>0$ are:

${\mathcal{I}}_{\xi }^{{\omega }_1}=\frac{1}{\mathrm{\aleph }!}{\int }_{{\omega }_1}^{\mathit{\gamma }}(\mathit{\gamma }-\mathit{\kappa }{)}^{\mathrm{\aleph }}(\mathit{\kappa }-{\omega }_1{)}^{\mathit{\beta }-1}\mathit{\varphi }(\mathit{\kappa })\mathit{d\kappa },\mathit{\gamma }>{\omega }_2,$

and

${ }^{{\omega }_2}{\mathcal{I}}_{\xi }=\frac{1}{\mathrm{\aleph }!}{\int }_{\gamma }^{{\omega }_2}(\mathit{\kappa }-\mathit{\gamma }{)}^{\mathrm{\aleph }}({\omega }_2-\mathit{\kappa }{)}^{\mathit{\beta }-1}\mathit{\varphi }(\mathit{\kappa })\mathit{d\kappa },\mathit{\gamma }<{\omega }_2,$

respectively. If one takes $\mathit{\xi }=\mathrm{\aleph }+1$ in this definition (for the left and right Conformable fractional integrals), one has the left and right Riemann–Liouville fractional integrals for $\mathit{\xi }\in \mathit{N}$.

The primary objective of this analysis is to utilize fractional calculus to derive novel Mercer type inequalities for harmonically convex mappings. The study will introduce the concept of the line of support to obtain a new fractional (H–H) Mercer type inequality. Additionally, we aim to derive additional trapezoid-type inequalities by utilizing the right and left Conformable fractional integral in the Mercer sense. Furthermore, when $\mathit{\xi }=\mathrm{\aleph }+1$ and $\mathit{\xi }=1$ in the obtained results, we will deduce inequalities of the classical Mercer type along with their different variants. Finally, to validate the reported results, we will compare them with diminished outcomes and conduct relevant implementations.

2. New Fractional H-H Mercer Type Inequalities for $\mathit{\varphi }\in \mathit{HC}(\mathbb{I})$

In this section, we present new conformable fractional (H–H) Mercer type inequalities that are derived using only the right or left fractional integrals, separately, for harmonically convex functions.

Theorem 2.1.

Let $\mathit{\varphi }:\mathbb{I}=[{\mathrm{\ell }}_1,{\mathrm{\ell }}_2]\subseteq (0,\mathrm{\infty })\to \mathbb{R}$ be a function such that $\mathit{\varphi }\in \mathit{L}[{\mathrm{\ell }}_1,{\mathrm{\ell }}_2]$ with $0<{\mathrm{\ell }}_1<{\mathrm{\ell }}_2$. If $\mathit{\varphi }\in \mathit{HC}(\mathbb{I})$ on the interval $\mathbb{I}=[{\mathrm{\ell }}_1,{\mathrm{\ell }}_2]$, then

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}{\mathcal{I}}_{\xi }^{\frac{1}{{\mathrm{\ell }}_2}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\mathrm{\ell }}_1})$

(2.1) $\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\mathit{\varphi }\left(\frac{(\xi +1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}\right),$(2.1)

and

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)}})\le (\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{)}^{\xi }\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}.{}^{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}{\mathcal{I}}_{\xi }(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1})$

$\le \frac{(\mathrm{\aleph }+1)\varphi \left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)+(\mathit{\xi }-\mathrm{\aleph })\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}\right)}{\mathit{\xi }+1}$

(2.2) $\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\frac{(\mathrm{\aleph }+1)\varphi ({\mathrm{\ell }}_2)+(\mathit{\xi }-\mathrm{\aleph })\mathit{\varphi }({\mathrm{\ell }}_1)}{\mathit{\xi }+1},$(2.2)

for all ${\mathrm{\ell }}_1,{\mathrm{\ell }}_2\in [{\omega }_1,{\omega }_2]$ and $\mathfrak{g}(\mathit{\kappa })=\mathit{\varphi }(\frac{1}{\kappa })$, $\mathit{\kappa }\in [\frac{1}{{\omega }_2},\frac{1}{{\omega }_1}]$.

Proof.

Since $\mathit{\varphi }$ is harmonic convex on $[{\omega }_1,{\omega }_2]$ by using Remark 1.1, $\mathfrak{g}(\mathit{\kappa })=\mathit{\varphi }(\frac{1}{\kappa })$ is convex on $[\frac{1}{{\omega }_2},\frac{1}{{\omega }_1}]$. Hence, using Remark 1.1, there is at least one line of support.

(2.3) $\mathit{A}(\mathit{\kappa })=\mathfrak{g}({\kappa }_0)+\mathit{c}\left(\mathit{\kappa }-{\kappa }_0\right)\le \mathfrak{g}(\mathit{\kappa }),$(2.3)

we put ${\kappa }_0=(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)})$ and $\mathit{\kappa }=\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1}$ in (2.3).

$\mathit{A}(\mathit{\kappa })=\mathfrak{g}\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)}\right)\text{break}+\mathit{c}\left(\mathit{\kappa }-\frac{1}{{\omega }_1}-\frac{1}{{\omega }_2}+\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)}\right)\le \mathfrak{g}(\mathit{\kappa }),$

for all $\mathit{\kappa }\in [\frac{1}{{\omega }_2},\frac{1}{{\omega }_1}]$ and $\mathit{c}\in [\mathfrak{g}{ }_{-}^{\prime }(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}),\mathfrak{g}{ }_{+}^{\prime }(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2})]$.

$\Rightarrow \mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})$

$+\mathit{c}(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1}-\frac{1}{{\omega }_1}-\frac{1}{{\omega }_2}+\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2})$

(2.4) $\le \mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1}}\right).$(2.4)

Multiplying all sides of (2.4) with $\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}$ and integrating over $[0,1]$ respect to $\mathit{\gamma }$, we have

${\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}[\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})+\mathit{c}(-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1}+\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2})]\mathit{d\gamma }+\frac{{\mathrm{\ell }}_1}{\mathrm{\aleph }!}[{\int }_0^1{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}(-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1})\mathit{d\gamma }+{\int }_0^1{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}(\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2})\mathit{d\gamma }]$

$=\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}+\frac{{\mathrm{\ell }}_1}{\mathrm{\aleph }!}[-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}$

$+\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}]$

(2.5) $\Rightarrow \mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}\le {\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{\varphi }(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1})\mathit{d\gamma }.$(2.5)

Using Mercer’s inequality, we have

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}$

(2.6) $\le {\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\left[\mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\mathit{\gamma \varphi }({\mathrm{\ell }}_2)-(1-\mathit{\gamma })\mathit{\varphi }({\mathrm{\ell }}_1)\right]\mathit{d\gamma }.$(2.6)

Since $\mathit{\varphi }$ is harmonic convex, we have $-(\mathit{\gamma \varphi }({\mathrm{\ell }}_2)+(1-\mathit{\gamma })\mathit{\varphi }({\mathrm{\ell }}_1))$ $\le -\mathit{\varphi }\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\mathit{\gamma }{\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2}\right)$ and (2.6) becomes

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}$

(2.7) $\le \frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}[\mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)]-{\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{\varphi }\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\mathit{\gamma }{\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2}\right)\mathit{d\gamma }.$(2.7)

Substitute $\frac{1}{\mathfrak{u}}=\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\mathit{\gamma }{\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2}$ in (2.7), we obtain

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}$

$\le \frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}[\mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)]-{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}\frac{1}{\mathrm{\aleph }!}{\int }_{\frac{1}{{\mathrm{\ell }}_2}}^{\frac{1}{{\mathrm{\ell }}_1}}{\left(\frac{1}{{\mathrm{\ell }}_1}-\mathfrak{u}\right)}^{\mathrm{\aleph }}{\left(\mathfrak{u}-\frac{1}{{\mathrm{\ell }}_2}\right)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{\varphi }\left(\frac{1}{\mathfrak{u}}\right)\mathit{d}\mathfrak{u}$

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}$

(2.8) $\le \frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}[\mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)]-{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}{\mathcal{I}}_{\xi }^{\frac{1}{{\mathrm{\ell }}_2}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\mathrm{\ell }}_1}).$(2.8)

It means that

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\frac{\mathrm{\aleph }!}{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}{\mathcal{I}}_{\xi }^{\frac{1}{{\mathrm{\ell }}_2}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\mathrm{\ell }}_1})$

(2.9) $=\mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}{\mathcal{I}}_{\xi }^{\frac{1}{{\mathrm{\ell }}_2}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\mathrm{\ell }}_1}).$(2.9)

Now for second inequality of (2.1). Put ${\kappa }_0=\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}$ and $\mathit{\kappa }=\frac{\gamma {\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}$ in (2.3) we get:

(2.10) $\mathit{\varphi }\left(\frac{(\xi +1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}\right)+\mathit{c}\left[\frac{\gamma {\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}\right]\le \mathit{\varphi }\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\mathit{\gamma }{\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2}\right),$(2.10)

for all $\mathit{\gamma }\in [0,1]$. Multiplying all sides of (2.10) with $\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}$ $\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}$ and integrating over $[0,1]$ respect to $\mathit{\gamma }$, we have:

$\mathit{\varphi }\left(\frac{(\xi +1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}\right)\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}\le {\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{\varphi }\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\mathit{\gamma }{\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2}\right)\mathit{d\gamma },$

put $\frac{1}{\mathfrak{u}}=\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\mathit{\gamma }{\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2}$, we obtain

$\mathit{\varphi }\left(\frac{(\xi +1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}\right)\le \frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}{\mathcal{I}}_{\xi }^{\frac{1}{{\mathrm{\ell }}_2}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\mathrm{\ell }}_1})$

(2.11) $-\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}{\mathcal{I}}_{\xi }^{\frac{1}{{\mathrm{\ell }}_2}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\mathrm{\ell }}_1})\le -\mathit{\varphi }\left(\frac{(\xi +1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}\right).$(2.11)

Adding $\mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)$ on both sides of (2.11), we have

$\mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}{\mathcal{I}}_{\xi }^{\frac{1}{{\mathrm{\ell }}_2}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\mathrm{\ell }}_1})$

(2.12) $\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\mathit{\varphi }\left(\frac{(\xi +1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}\right),$(2.12)

and on combining (2.9) and (2.12), we obtain (2.1). This completes the proof.

Now we prove (2.2). Let $\mathit{\kappa }=\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1}\Rightarrow \mathit{\gamma }=\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\left[\mathit{\kappa }-(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1})\right]$ and (2.5) becomes:

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{(\mathit{\xi }+1){\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}$

$\le {\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{\varphi }(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1})\mathit{d\gamma }$

$=(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{)}^{\xi }}^{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}{\mathcal{I}}_{\xi }(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1})$

$\Rightarrow \mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{(\mathrm{\aleph }+1){\mathrm{\ell }}_1+(\mathit{\xi }-\mathrm{\aleph }){\mathrm{\ell }}_2}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)}})$

(2.13) $\le (\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{)}^{\xi }\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}{}^{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}{\mathcal{I}}_{\xi }(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}).$(2.13)

Now we prove other two inequalities of (2.2). By using the harmonic convexity of $\mathit{\varphi }$ on $[{\omega }_1,{\omega }_2]$, we have:

$\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1}}\right)=\mathit{\varphi }\left(\frac{1}{\gamma \left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}\right)+(1-\mathit{\gamma })\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}\right)}\right)$

$\Rightarrow \mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1}}\right)\le \mathit{\gamma \varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)+(1-\mathit{\gamma })\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}\right)$

$\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\mathit{\gamma \varphi }({\mathrm{\ell }}_2)-(1-\mathit{\gamma })\mathit{\varphi }({\mathrm{\ell }}_1).$

Multiplying the above inequality with $\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}$ and integrating over $[0,1]$ respect to $\mathit{\gamma }$, we have

${\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1}}\right)\mathit{d\gamma }$

$\le \mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right){\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }+1}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{d\gamma }+\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}\right){\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }}\mathit{d\gamma }$

$\le {\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}[\mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)]-\mathit{\varphi }({\mathrm{\ell }}_2){\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }+1}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{d\gamma }$

$-\mathit{\varphi }({\mathrm{\ell }}_1){\int }_0^1\frac{1}{\mathrm{\aleph }!}{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }}\mathit{d\gamma }$

$\Rightarrow \mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_2}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_1}}\right)\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}$

$\le \frac{(\mathrm{\aleph }+1)\varphi \left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)+(\mathit{\xi }-\mathrm{\aleph })\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}\right)}{\mathit{\xi }+1}\frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}$

(2.14) $\le \frac{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathrm{\aleph }!}\left[\mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\frac{(\mathrm{\aleph }+1)\varphi ({\mathrm{\ell }}_2)+(\mathit{\xi }-\mathrm{\aleph })\mathit{\varphi }({\mathrm{\ell }}_1)}{\mathit{\xi }+1}\right].$(2.14)

By changing variable, (2.14) becomes

$\Rightarrow {(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1})}^{\mathit{\xi }\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}{\mathcal{I}}_{\xi }(\mathit{\varphi }\circ \mathfrak{g})\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}\right)$

$\le \frac{(\mathrm{\aleph }+1)\varphi \left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)+(\mathit{\xi }-\mathrm{\aleph })\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}\right)}{\mathit{\xi }+1}$

(2.15) $\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\frac{(\mathrm{\aleph }+1)\varphi ({\mathrm{\ell }}_2)+(\mathit{\xi }-\mathrm{\aleph })\mathit{\varphi }({\mathrm{\ell }}_1)}{\mathit{\xi }+1}.$(2.15)

By combining (2.13) and (2.15), we obtain (2.2).□

Remark 2.1. If we put $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ in Theorem 2.1, we have

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{{\mathrm{\ell }}_1+{\mathrm{\ell }}_2}{2{\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-{\int }_0^1\mathit{\varphi }\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\mathit{\gamma }{\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2}\right)\mathit{d\gamma }$

(2.16) $\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\mathit{\varphi }(\frac{2{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_1+{\mathrm{\ell }}_2}),$(2.16)

and

$\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{{\mathrm{\ell }}_1+{\mathrm{\ell }}_2}{2{\mathrm{\ell }}_1{\mathrm{\ell }}_2}})\le \frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{\int }_{\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}}^{\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}}\frac{\mathit{\varphi }(\mathit{\kappa })}{{\kappa }^2}\mathit{d\kappa }$

(2.17) $\le \frac{\varphi \left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)+\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}\right)}{2}\le \mathit{\varphi }({\omega }_1)+\mathit{\varphi }({\omega }_2)-\frac{\varphi ({\mathrm{\ell }}_1)+\mathit{\varphi }({\mathrm{\ell }}_2)}{2},$(2.17)

for all ${\mathrm{\ell }}_1,{\mathrm{\ell }}_2\in [{\omega }_1,{\omega }_2]$. The proof of inequality (2.16) is given by Baloch et al. in [(Baloch et al. 2020), Theorem 3.5]. Inequality (2.17) is better than the inequality proved by Baloch et al. in [(Baloch et al. 2021), Theorem 2.1]. As (2.17) is refinement of that inequality.

Remark 2.2. If we put $\mathit{\xi }=1,{\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$ in (2.2), we have (1.3) (H-H inequality) appeared in (Butt et al. 2022).

3. Novel Mercer type inequalities for Ф є HC($\mathbb{I}$) via conformable fractional integrals

We make the following assumptions throughout the rest of this paper:

$\mathrm{\Omega }=$ Let $\mathit{\varphi }:\mathit{I}=[{\omega }_1,{\omega }_2]\subseteq (0,\mathrm{\infty })\to \mathbb{R}$ be a differentiable function on $[{\omega }_1,{\omega }_2]$ with $0<{\omega }_1\le {\omega }_2$.

$I_{{\varphi }_1}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)=\frac{(\xi -\mathrm{\aleph })\varphi (\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}})+(\mathrm{\aleph }+1)\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})}{\mathit{\xi }+1}-(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{)}^{\xi }\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}{\mathcal{I}}_{\xi }^{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}),$

$I_{{\varphi }_2}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)=\frac{(\xi -\mathrm{\aleph })\varphi (\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})+(\mathrm{\aleph }+1)\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}})}{\mathit{\xi }+1}-{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}\text{break}{\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}}^{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}{\mathcal{I}}_{\xi }(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}),$

with $\mathrm{\aleph }=0,1,2,3,\dots$ and $\mathit{\xi }=(\mathrm{\aleph },\mathrm{\aleph }+1]$.

In this section, we prove two new Conformable fractional equalities used in forward results.

Lemma 3.1.

If $\mathit{\varphi }{ }^{\mathrm{\prime }}\in \mathcal{L}[{\mathrm{\ell }}_1,{\mathrm{\ell }}_2]$ along with assumption $\mathrm{\Omega }$, then following identity for fractional integral holds:

$I_{{\varphi }_1}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)=\frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\int }_0^1\left[\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-(\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\\ \mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right]$

(3.1) $\times \frac{1}{{(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_1}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_2})}^2}\mathit{\varphi }{ }^{\mathrm{\prime }}(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_1}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_2}})\mathit{d\gamma },$(3.1)

with $\mathrm{\aleph }=0,1,2,\dots$ and $\mathit{\xi }=(\mathrm{\aleph },\mathrm{\aleph }+1]$.

Proof.

Let $A_{\gamma }=\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{\gamma }{{\mathrm{\ell }}_1}-\frac{(1-\mathit{\gamma })}{{\mathrm{\ell }}_2}$. Applying partial integration to the right-hand side of equation (3.1) as follows:

$=\frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)}{\int }_0^1\left[(\mathit{\xi }+1)\frac{{\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}-(\mathit{\xi }-\mathrm{\aleph })\right]\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{\varphi }{ }^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma }$

$=\left[\frac{1}{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\int }_0^1\left({\int }_0^{\gamma }{\kappa }^{\mathrm{\aleph }}(1-\mathit{\kappa }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{d\kappa }\right)\frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2{\left(A_{\gamma }\right)}^2}\mathit{\varphi }{ }^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma }-\frac{\mathit{\xi }-\mathrm{\aleph }}{\mathit{\xi }+1}\mathit{\varphi }({\frac{1}{A}}_{\mathit{\gamma }}){|}_0^1\right]$

$=[\frac{1}{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}\left({\int }_0^{\gamma }{\kappa }^{\mathrm{\aleph }}(1-\mathit{\kappa }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{d\kappa })\mathit{\varphi }\left({\frac{1}{A}}_{\mathit{\gamma }}\right){|}_0^1-{\int }_0^1{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{\varphi }({\frac{1}{A}}_{\mathit{\gamma }}\right)\mathit{d\gamma }$

$-\frac{\mathit{\xi }-\mathrm{\aleph }}{\mathit{\xi }+1}\left(\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})-\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)\right)\right]$

$=\left[\frac{1}{\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}(\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})-{\int }_0^1{\gamma }^{\mathrm{\aleph }}(1-\mathit{\gamma }{)}^{\mathit{\xi }-\mathrm{\aleph }-1}\mathit{\varphi }({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma }\right)$

$+\frac{\mathit{\xi }-\mathrm{\aleph }}{\mathit{\xi }+1}\left(\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)-\mathit{\varphi }\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})\right)\right]$

$=\frac{(\xi -\mathrm{\aleph })\varphi (\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}})+(\mathrm{\aleph }+1)\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})}{\mathit{\xi }+1}-(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{)}^{\xi }\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}{\mathcal{I}}_{\xi }^{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}).$

This completes the proof.□

Remark 3.1. Another approach to proof Lemma 3.1 is by selecting $\mathfrak{a}=\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}$ and $\mathfrak{b}=\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}$ in Lemma 3.2 of (Sanli et al. 2018). This choice directly aligns Lemma 3.2 of (Sanli et al. 2018) with (3.2).

Remark 3.2. If we take $\mathit{\xi }=\mathrm{\aleph }+1$ in Lemma 3.1, then we have

$\frac{\varphi (\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}})+\mathit{\xi \varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})}{\mathit{\xi }+1}-\frac{1}{\mathit{\beta }(\mathit{\xi },1)}{\int }_0^1{\gamma }^{\mathit{\xi }-1}\mathit{\varphi }({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma }=\frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathit{\xi },1)}$

${\int }_0^1\left[\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathit{\xi },1)-& \mathit{\hspace{0.17em}}\\ \mathit{\beta }(\mathit{\xi },1)& \mathit{\hspace{0.17em}}\end{array}\right]\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{\varphi }{ }^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma },$

which is new in the literature.

Remark 3.3. If we take ${\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$ in Lemma 3.1, then we have

$\frac{(\xi -\mathrm{\aleph })\varphi ({\mathrm{\ell }}_1)+(\mathrm{\aleph }+1)\mathit{\varphi }({\mathrm{\ell }}_2)}{\mathit{\xi }+1}-(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{)}^{\xi }\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}{\mathcal{I}}_{\xi }^{\frac{1}{{\mathrm{\ell }}_2}}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\mathrm{\ell }}_1})$

$=\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}{(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\int }_0^1\left[\begin{array}{c}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })\end{array}\right]\mathit{\hspace{0.17em}}\times \frac{1}{{\left(\gamma {\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2\right)}^2}\mathit{\varphi }\mathrm{\prime }(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\left(\mathit{\gamma }{\mathrm{\ell }}_1+(1-\mathit{\gamma }){\mathrm{\ell }}_2\right)})\mathit{d\gamma },$which is appeared in (Sanli et al. 2018).

Remark 3.4. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2$ and $\mathit{\xi }=\mathrm{\aleph }+1$ in Lemma 3.1, then we have

$\frac{\varphi ({\mathrm{\ell }}_1)+\mathit{\xi \varphi }({\mathrm{\ell }}_2)}{\mathit{\xi }+1}-\mathrm{\Gamma }(\mathit{\xi }+1){\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}{\mathbb{I}}_{\frac{1}{{\mathrm{\ell }}_2}+}^{\mathit{\xi }}(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\mathrm{\ell }}_1})$

$=\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}{\mathit{\xi }+1}{\int }_0^1\frac{(\xi +1)(1-\gamma {)}^{\xi }-1}{{(\gamma {\mathrm{\ell }}_2+(1-\mathit{\gamma }){\mathrm{\ell }}_1)}^2}{\varphi }^{\prime }\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\mathit{\gamma }{\mathrm{\ell }}_2+(1-\mathit{\gamma }){\mathrm{\ell }}_1}\right)\mathit{d\gamma },$

which is appeared in (Sanli et al. 2020).

Remark 3.5. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2,\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ in Lemma 3.1, then we have

$\frac{\varphi ({\mathrm{\ell }}_1)+\mathit{\varphi }({\mathrm{\ell }}_2)}{2}-\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{\int }_{{\mathrm{\ell }}_1}^{{\mathrm{\ell }}_2}\frac{\mathit{\varphi }(\mathit{\kappa })}{{\kappa }^2}\mathit{d\kappa }\text{break}=\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}{2}{\int }_0^1\frac{1-2\mathit{\gamma }}{{(\gamma {\mathrm{\ell }}_2+(1-\mathit{\gamma }){\mathrm{\ell }}_1)}^2}{\varphi }^{\prime }\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{\mathit{\gamma }{\mathrm{\ell }}_2+(1-\mathit{\gamma }){\mathrm{\ell }}_1}\right)\mathit{d\gamma },$

which is appeared in (ͺͺCan 2014).

Lemma 3.2.

If $\mathit{\varphi }{ }^{\mathrm{\prime }}\in \mathcal{L}[{\mathrm{\ell }}_1,{\mathrm{\ell }}_2]$ along with assumption $\mathrm{\Omega }$, then following identity for fractional integral holds:

$\frac{(\xi -\mathrm{\aleph })\varphi (\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})+(\mathrm{\aleph }+1)\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}})}{\mathit{\xi }+1}-{\left(\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}\right)}^{\mathit{\xi }}{\frac{\mathrm{\Gamma }(\mathit{\xi }+1)}{\mathrm{\Gamma }(\mathit{\xi }-\mathrm{\aleph })}}^{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}{\mathcal{I}}_{\xi }(\mathit{\varphi }\circ \mathfrak{g})(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1})$

(3.2) $=\frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\int }_0^1\left[\begin{array}{cc}(\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }+1){\beta }_{1-\mathit{\gamma }}(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right]\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{\varphi }{ }^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma },$(3.2)

with $\mathrm{\aleph }=0,1,2,3,\dots$ and $\mathit{\xi }=(\mathrm{\aleph },\mathrm{\aleph }+1]$.

Proof.

In Lemma 3.2 of Sanli and Köroğlu (2018), if we choose

$\mathfrak{a}=\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}},$

and

$\mathfrak{b}=\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}},$

then Lemma 3.2 of (Sanli et al. 2018) reduces to (3.2).□

Remark 3.6. If we take $\mathit{\xi }=\mathrm{\aleph }+1$ in Lemma 3.2, then we have

$\frac{\varphi (\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})+\mathit{\xi \varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}})}{\mathit{\xi }+1}-\frac{1}{\mathit{\beta }(\mathit{\xi },1)}{\int }_0^1(1-\mathit{\gamma }{)}^{\mathit{\xi }-1}\mathit{\varphi }({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma }=\frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathit{\xi },1)}$

${\int }_0^1\left[\begin{array}{cc}\mathit{\beta }(\mathit{\xi },1)-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }+1){\beta }_{1-\mathit{\gamma }}(\mathit{\xi },1)& \mathit{\hspace{0.17em}}\end{array}\right]\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{\varphi }{ }^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma },$

which is new in literature.

Remark 3.7. If we take ${\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$ in Lemma 3.2, then we have the equality proved in (Sanli et al. 2018).

Remark 3.8. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2$ and $\mathit{\xi }=\mathrm{\aleph }+1$ in Lemma 3.2, then we have the equality proved in (Sanli et al. 2020).

Remark 3.9. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2,\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ in Lemma 3.1, then we have the equality proved in (ÍşCan 2014).

Theorem 3.1.

If $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex function for some fixed $\mathit{q}\ge 1$ and $\mathit{\varphi }{ }^{\mathrm{\prime }}\in \mathcal{L}[{\omega }_1,{\omega }_2]$ along with assumption $\mathrm{\Omega }$ and $0<\mathit{\xi }<1$, then the following inequality for fractional integral holds:

$|I_{{\varphi }_1}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|$

(3.3) $\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{K_1}^{1-\frac{1}{q}}({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi }){\left[\begin{array}{cc}{K}_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-K_3({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}},$(3.3)

where

$K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })={\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{d\gamma }$

$K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })={\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{\gamma d\gamma }$

$K_3({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })=K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })-K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi }),$

with $\mathrm{\aleph }=0,1,2,3,\dots$ and $\mathit{\xi }=(\mathrm{\aleph },\mathrm{\aleph }+1]$.

Proof.

Using Lemma 3.1, Power-mean inequality and since $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex, by using Mercer inequality, we have

$|I_{{\varphi }_1}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|$

$\times \frac{1}{{\left(A_{\gamma }\right)}^2}\left|\mathit{\varphi }{ }^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\right|\mathit{d\gamma }$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\left(\begin{array}{cc}{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|& \mathit{\hspace{0.17em}}\\ \times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{d\gamma }& \mathit{\hspace{0.17em}}\end{array}\right)}^{1-\frac{1}{q}}$

$\times {\left(\begin{array}{cc}{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|& \mathit{\hspace{0.17em}}\\ \times \frac{1}{{\left(A_{\gamma }\right)}^2}{\left|\mathit{\varphi }{\mathit{ }}^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\right|}^{\mathit{q}}\mathit{d\gamma }& \mathit{\hspace{0.17em}}\end{array}\right)}^{\frac{1}{q}}$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\left(\begin{array}{cc}{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|& \mathit{\hspace{0.17em}}\\ \times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{d\gamma }& \mathit{\hspace{0.17em}}\end{array}\right)}^{1-\frac{1}{q}}$

$\times {\left(\begin{array}{cc}{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|& \mathit{\hspace{0.17em}}\\ \times \frac{1}{{\left(A_{\gamma }\right)}^2}\left[|{\varphi }^{\prime }({\omega }_1{)|}^q+|{\varphi }^{\prime }({\omega }_2{)|}^q-\mathit{\gamma }|{\varphi }^{\prime }({\mathrm{\ell }}_1{)|}^q-(1-\mathit{\gamma })|{\varphi }^{\prime }({\mathrm{\ell }}_2{)|}^q\right]\mathit{d\gamma }& \mathit{\hspace{0.17em}}\end{array}\right)}^{\frac{1}{q}}$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\left(\begin{array}{cc}{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|& \mathit{\hspace{0.17em}}\\ \times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{d\gamma }& \mathit{\hspace{0.17em}}\end{array}\right)}^{1-\frac{1}{q}}$

$\times {\left(\begin{array}{cccc}\left[|{\varphi }^{\prime }({\omega }_1{)|}^q+|{\varphi }^{\prime }({\omega }_2{)|}^q\right]{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{d\gamma }& \mathit{\hspace{0.17em}}& \mathit{\hspace{0.17em}}& \mathit{\hspace{0.17em}}\\ -|{\varphi }^{\prime }({\mathrm{\ell }}_1{)|}^q{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{\gamma d\gamma }& \mathit{\hspace{0.17em}}& \mathit{\hspace{0.17em}}& \mathit{\hspace{0.17em}}\\ -|{\varphi }^{\prime }({\mathrm{\ell }}_2{)|}^q{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|\times \frac{1}{{\left(A_{\gamma }\right)}^2}(1-\mathit{\gamma })\mathit{d\gamma }& \mathit{\hspace{0.17em}}& \mathit{\hspace{0.17em}}& \mathit{\hspace{0.17em}}\end{array}\right)}^{\frac{1}{q}}$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{K_1}^{1-\frac{1}{q}}({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi }){\left[\begin{array}{cc}{K}_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-K_3({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}}.$

This completes the proof.□

Remark 3.10. If we take $\mathit{\xi }=\mathrm{\aleph }+1$ in Theorem 3.1, then we have

$\left|\frac{\varphi (\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}})+\mathit{\xi \varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})}{\mathit{\xi }+1}-\frac{1}{\mathit{\beta }(\mathit{\xi },1)}{\int }_0^1{\gamma }^{\mathit{\xi }-1}\mathit{\varphi }({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma }\right|$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{K_1}^{1-\frac{1}{q}}({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi }){\left[\begin{array}{cc}{K}_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-K_3({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}},$

where

$K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })={\int }_0^1\left[\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathit{\xi },1)-& \mathit{\hspace{0.17em}}\\ \mathit{\beta }(\mathit{\xi },1)& \mathit{\hspace{0.17em}}\end{array}\right]\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{d\gamma }$

$K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })={\int }_0^1\left[\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathit{\xi },1)-& \mathit{\hspace{0.17em}}\\ \mathit{\beta }(\mathit{\xi },1)& \mathit{\hspace{0.17em}}\end{array}\right]\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{\gamma d\gamma }$

$K_3({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })=K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })-K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi }),$

which is new in the literature.

Remark 3.11. If we take $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ in Theorem 3.1, then we have

$\left|\frac{\varphi (\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}})+\mathit{\varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})}{2}-{\int }_0^1\mathit{\varphi }({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma }\right|$

$\le \frac{2{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{K_1}^{1-\frac{1}{q}}({\mathrm{\ell }}_1,{\mathrm{\ell }}_2){\left[\begin{array}{cc}{K}_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-K_3({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}},$

where

$K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)={\int }_0^1\times \frac{|1-2\gamma |}{{\left(A_{\gamma }\right)}^2}\mathit{d\gamma }$

$K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)={\int }_0^1\times \frac{|1-2\gamma |}{{\left(A_{\gamma }\right)}^2}\mathit{\gamma d\gamma }$

$K_3({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)=K_1({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)-K_2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2),$

which is new in the literature.

Remark 3.12. If we take ${\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$ in Theorem 3.1, then we have the inequality proved in (Sanli et al. 2018).

Remark 3.13. If we take ${\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$ and $\mathit{\xi }=\mathrm{\aleph }+1$ in Theorem 3.1, then we have the inequality proved in (Sanli et al. 2020), Theorem 9].

Remark 3.14. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2,\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ in Theorem 3.1, then we have then we have the inequality proved in (ÍşCan 2014).

Example 3.1. Case 1: Let $\mathit{\varphi }(\mathit{\kappa })=\frac{{\kappa }^4}{4},\mathit{\kappa }>0$. If we set $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$, ${\omega }_1={\mathrm{\ell }}_1$, ${\omega }_1={\mathrm{\ell }}_2$, ${\mathrm{\ell }}_1=2$, ${\mathrm{\ell }}_2=5$ and $\mathit{q}\in [1.1,100]$, then by Proposition 1.1 mapping, $|\mathit{\varphi }{ }^{\mathrm{\prime }}(\mathit{\kappa }){|}^q={\kappa }^{3\mathit{q}}$ is harmonic convex, then we can say that the inequality (3.3) will deduce to

$-15[0.055{]}^{\frac{\mathit{q}-1}{q}}\times \{(0.016)\times [2{]}^{3\mathit{q}}+(0.039)\times [5{]}^{3\mathit{q}}{\}}^{\frac{1}{q}}$

$\le \frac{2^4+5^4}{8}-\frac{5}{6}{\int }_2^5{\mathfrak{u}}^2\mathit{d}\mathfrak{u}\approx 47.625$

$\le 15[0.055{]}^{\frac{\mathit{q}-1}{q}}\times \{(0.016)\times [2{]}^{3\mathit{q}}+(0.039)\times [5{]}^{3\mathit{q}}{\}}^{\frac{1}{q}}.$

Case 2: Let $\mathit{\varphi }(\mathit{\kappa })=\frac{{\kappa }^4}{4},\mathit{\kappa }>0$. Suppose we set $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$, ${\omega }_1=2$, ${\omega }_2=5$, $\mathit{q}=2$ and ${\mathrm{\ell }}_1=[2,3]$, ${\mathrm{\ell }}_2=[4,5]$ in (3.3).

Figure 1. Case 1 visual illustration for $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$, ${\omega }_1={\mathrm{\ell }}_1$, ${\omega }_2={\mathrm{\ell }}_2$, ${\mathrm{\ell }}_1=2$, ${\mathrm{\ell }}_2=5$ and $\mathit{q}\in [1.1,100]$.

Figure 2. Case 2 visual illustration for $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$, ${\omega }_1=2$, ${\omega }_2=5$, $\mathit{q}=2$ and ${\mathrm{\ell }}_1=[2,3]$, ${\mathrm{\ell }}_2=[4,5]$.

Left, middle, and right mappings from the inequalities (3.3) are plotted against $\mathit{q}\in [1.1,100]$ in Figure 1. Left, middle, and right mappings from the inequalities (3.3) are plotted against ${\mathrm{\ell }}_1=[1,1.5]$, ${\mathrm{\ell }}_2=[4,5]$ in Figure 2. The graphs of the functions prove the correctness of the Theorem 3.1 with $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$.

Theorem 3.2.

If $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex function for some fixed $\mathit{q}\ge 1$ and $\mathit{\varphi }{ }^{\mathrm{\prime }}\in \mathcal{L}[{\omega }_1,{\omega }_2]$ along with assumption $\mathrm{\Omega }$ and $0<\mathit{\xi }<1$, then the following inequality for fractional integral holds:

$|I_{{\varphi }_1}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|$

(3.4) $\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{K_4}^{\frac{1}{p}}(\mathit{\xi },\mathrm{\aleph }){\left[\begin{array}{cc}{K}_5(\mathit{\xi },\mathrm{\aleph })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_5(\mathit{\xi },\mathrm{\aleph })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_6({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q-K_7({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}},$(3.4)

where

$K_4(\mathit{\xi },\mathrm{\aleph })={\int }_0^1{\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma },$

$K_5({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })=[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},1;2,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right),$

$K_6({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)={\frac{{(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2})}^{-2\mathit{q}}}{2}}_2{F}_1\left(2\mathit{q},2;3;1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right),$

$K_7({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)={\frac{{(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2})}^{-2\mathit{q}}}{2}}_2{F}_1\left(2\mathit{q},1;3;1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right).$

Here, $\frac{1}{p}+\frac{1}{q}=1$, $\mathrm{\aleph }=0,1,2,3,\dots$ and $\mathit{\xi }=(\mathrm{\aleph },\mathrm{\aleph }+1]$.

Proof.

By using Lemma 3.1, Hölder’s inequality and since $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex, by using Mercer inequality, we have

$|I_{{\varphi }_1}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }+1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|$

$\times \frac{1}{{\left(A_{\gamma }\right)}^2}\left|\mathit{\varphi }{ }^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\right|\mathit{d\gamma }$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\left({\int }_0^1{\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma }\right)}^{\frac{1}{p}}$

$({\int }_0^1\frac{1}{{\left(A_{\gamma }\right)}^{2\mathit{q}}}{\left|\mathit{\varphi }{\mathit{ }}^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\right|}^{\mathit{q}}\mathit{d\gamma }{)}^{\frac{1}{q}}$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\left({\int }_0^1{\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma }\right)}^{\frac{1}{p}}$

$({\int }_0^1\frac{1}{{\left(A_{\gamma }\right)}^{2\mathit{q}}}\left[\begin{array}{cc}|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -\mathit{\gamma }|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-(1-\mathit{\gamma })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]\mathit{d\gamma }{)}^{\frac{1}{q}}$

(3.5) $\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{K_4}^{\frac{1}{p}}({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })[\begin{array}{cc}{K}_5(\mathit{\xi },\mathrm{\aleph })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_5(\mathit{\xi },\mathrm{\aleph })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_6({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q-K_7({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q& \mathit{\hspace{0.17em}}\end{array}{]}^{\frac{1}{q}}.$(3.5)

Calculating the appearing integrals in (3.5), we have

${\int }_0^1\frac{1}{{\left(A_{\gamma }\right)}^{2\mathit{q}}}\mathit{d\gamma }={\int }_0^1\frac{1}{{\left[\gamma \left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}\right)+(1-\mathit{\gamma })\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}\right)\right]}^{2\mathit{q}}}\mathit{d\gamma }$

$={\int }_0^1\frac{1}{{\left[\gamma \left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}\right)+(1-\mathit{\gamma })\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}\right)\right]}^{2\mathit{q}}}\mathit{d\gamma }$

$=[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}{\int }_0^1(1-\mathit{\gamma }\left(1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right){)}^{-2\mathit{q}}\mathit{d\gamma }$

$=[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},1;2,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)$

$=K_5({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })$

${\int }_0^1\frac{\gamma }{{\left(A_{\gamma }\right)}^{2\mathit{q}}}\mathit{d\gamma }=\frac{{(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2})}^{-2\mathit{q}}}{2}{\int }_0^1\mathit{\gamma }\left[1-\mathit{\gamma }\left(\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)\right]\mathit{d\gamma }$

$={\frac{{(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2})}^{-2\mathit{q}}}{2}}_2{F}_1\left(2\mathit{q},2;3;1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)$

$=K_6({\mathrm{\ell }}_1,{\mathrm{\ell }}_2),$

and

${\int }_0^1\frac{1-\mathit{\gamma }}{{\left(A_{\gamma }\right)}^{2\mathit{q}}}\mathit{d\gamma }=\frac{{(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2})}^{-2\mathit{q}}}{2}{\int }_0^1(1-\mathit{\gamma })\left[1-\mathit{\gamma }\left(\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)\right]\mathit{d\gamma }$

$={\frac{{(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2})}^{-2\mathit{q}}}{2}}_2{F}_1\left(2\mathit{q},1;3;1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)$

$=K_7({\mathrm{\ell }}_1,{\mathrm{\ell }}_2).$

This ends the proof.□

Remark 3.15. If we take $\mathit{\xi }=\mathrm{\aleph }+1$ in Theorem 3.2, then we have

$\frac{\varphi (\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}})+\mathit{\xi \varphi }(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}})}{\mathit{\xi }+1}-\frac{1}{\mathit{\beta }(\mathit{\xi },1)}{\int }_0^1{\gamma }^{\mathit{\xi }-1}\mathit{\varphi }({\frac{1}{A}}_{\mathit{\gamma }})\mathit{d\gamma }\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathit{\xi },1)}$

${\int }_0^1\frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathit{\xi },1)}{K_4}^{\frac{1}{p}}(\mathit{\xi }){\left[\begin{array}{cc}{K}_5(\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_5(\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_6({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q-K_7({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}},$

where

$K_4(\mathit{\xi })={\int }_0^1{\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathit{\xi },1)-& \mathit{\hspace{0.17em}}\\ \mathit{\beta }(\mathit{\xi },1)& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma },$

and $K_5-K_7$ are defined above. This is new inequality in the literature.

Remark 3.16. If we take ${\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$ in Theorem 3.2, then we have the inequality proved in (Sanli et al. 2018).

Remark 3.17. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2$ and $\mathit{\xi }=\mathrm{\aleph }+1$ in Theorem 3.2, then we have the inequality proved in (Sanli et al. 2020), Theorem 10].

Remark 3.18. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2,\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ in Theorem 3.2, then we have then we have the inequality proved in (ÍşCan 2014).

Example 3.2. Case 1: Let $\mathit{\varphi }(\mathit{\kappa })=\frac{{\kappa }^4}{4},\mathit{\kappa }>0$. If we set $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$, ${\omega }_1={\mathrm{\ell }}_1$, ${\omega }_1={\mathrm{\ell }}_2$, ${\mathrm{\ell }}_1=1$, ${\mathrm{\ell }}_2=5$ and $\mathit{q}\in [1.1,100]$ in (3.4), then by Proposition 1.1 mapping, $|\mathit{\varphi }{ }^{\mathrm{\prime }}(\mathit{\kappa }){|}^q={\kappa }^{3\mathit{q}}$ is harmonic convex.

$-10[\frac{\mathit{q}-1}{2\mathit{q}-1}{]}^{\frac{\mathit{q}-1}{q}}\times {\left[{\mu }_3{(1)}^{3\mathit{q}}+{\mu }_4{(5)}^{3\mathit{q}}\right]}^{\frac{1}{q}}$

$\le 21-\frac{5}{4}{\int }_1^5\frac{\kappa }{3}\mathit{d\kappa }\approx 65.33$

(3.6) $\le 10[\frac{\mathit{q}-1}{2\mathit{q}-1}{]}^{\frac{\mathit{q}-1}{q}}\times {\left[{\mu }_3{(1)}^{3\mathit{q}}+{\mu }_4{(5)}^{3\mathit{q}}\right]}^{\frac{1}{q}},$(3.6)

where

${\mu }_3=\frac{\left[1^{2-2\mathit{q}}+5^{1-2\mathit{q}}[4(1-2\mathit{q})-1]\right]}{{2(4)}^2(1-\mathit{q})(1-2\mathit{q})}$

${\mu }_4=\frac{\left[5^{2-2\mathit{q}}+1^{1-2\mathit{q}}[4(1-2\mathit{q})-1]\right]}{{2(4)}^2(1-\mathit{q})(1-2\mathit{q})}.$

Case 2: Let $\mathit{\varphi }(\mathit{\kappa })=\frac{{\kappa }^4}{4},\mathit{\kappa }>0$. Suppose we set $\mathit{\xi }=1,{\omega }_1=1$, ${\omega }_2=5$, $\mathit{q}=2$ and ${\mathrm{\ell }}_1=[1,1.5]$, ${\mathrm{\ell }}_2=[4,5]$ in (3.4).

Figure 3. Case 1 visual illustration for $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$, ${\omega }_1={\mathrm{\ell }}_1$, ${\omega }_2={\mathrm{\ell }}_2$, ${\mathrm{\ell }}_1=1$, ${\mathrm{\ell }}_2=5$ and $\mathit{q}\in [1.1,100]$.

Figure 4. Case 2 visual illustration for $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1,{\omega }_1=1$, ${\omega }_2=5$, $\mathit{q}=2$ and ${\mathrm{\ell }}_1=[1,1.5]$, ${\mathrm{\ell }}_2=[4,5]$.

Left, middle, and right mappings from the inequalities (3.4) are plotted against $\mathit{q}\in [1.1,100]$ in Figure 3. Left, middle, and right mappings from the inequalities (3.4) are plotted against ${\mathrm{\ell }}_1=[1,1.5]$, ${\mathrm{\ell }}_2=[4,5]$ in Figure 4. The graphs of the functions prove the correctness of the Theorem 3.2 with $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$.

Theorem 3.3.

If $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex function for some fixed $\mathit{q}\ge 1$ and $\mathit{\varphi }{ }^{\mathrm{\prime }}\in \mathcal{L}[{\omega }_1,{\omega }_2]$ along with assumption $\mathrm{\Omega }$ and $0<\mathit{\xi }<1$, then the following inequality for fractional integral holds:

$|I_{{\varphi }_1}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}$

$\left\{{{\mathrm{{\rm Y}}}_1}^{\frac{1}{p}}{\left[\begin{array}{cc}{\mathrm{\Lambda }}_1|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+{\mathrm{\Lambda }}_1|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -{\mathrm{\Lambda }}_2|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-{\mathrm{\Lambda }}_3|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}}\right.$

$+{{\mathrm{{\rm Y}}}_2}^{\frac{1}{p}}{\left[\begin{array}{cc}{\mathrm{\Lambda }}_4|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+{\mathrm{\Lambda }}_4|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -{\mathrm{\Lambda }}_5|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-{\mathrm{\Lambda }}_2|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}}\},$

where

${\mathrm{{\rm Y}}}_1={\left({\int }_0^1(1-\mathit{\gamma }){\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma }\right)}^{\frac{1}{p}},$

${\mathrm{{\rm Y}}}_2={\left({\int }_0^1\mathit{\gamma }{\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma }\right)}^{\frac{1}{p}},$

${\mathrm{\Lambda }}_1=\frac{1}{2}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},1;3,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right),$

${\mathrm{\Lambda }}_2=\frac{1}{6}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},2;4,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right),$

${\mathrm{\Lambda }}_3=\frac{1}{3}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},1;4,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right),$

${\mathrm{\Lambda }}_4=\frac{1}{2}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},2;3,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right),$

${\mathrm{\Lambda }}_5=\frac{1}{3}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},3;4,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right).$

Here, $\frac{1}{p}+\frac{1}{q}=1$, $\mathrm{\aleph }=0,1,2,3,\dots$ and $\mathit{\xi }=(\mathrm{\aleph },\mathrm{\aleph }+1]$.

Proof.

By taking modulus in Lemma 3.1, using modified Hölder’s inequality [35] and since $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex, by using Mercer inequality, we have

$|I_{{\varphi }_1}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}\{{\left({\int }_0^1(1-\mathit{\gamma }){\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma }\right)}^{\frac{1}{p}}$

$({\int }_0^1\frac{1-\mathit{\gamma }}{{\left(A_{\gamma }\right)}^{2\mathit{q}}}{\left|\mathit{\varphi }{\mathit{ }}^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\right|}^{\mathit{q}}\mathit{d\gamma }{)}^{\frac{1}{q}}$

$+{\left({\int }_0^1\mathit{\gamma }{\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma }\right)}^{\frac{1}{p}}$

$({\int }_0^1\frac{\gamma }{{\left(A_{\gamma }\right)}^{2\mathit{q}}}{\left|\mathit{\varphi }{\mathit{ }}^{\mathrm{\prime }}({\frac{1}{A}}_{\mathit{\gamma }})\right|}^{\mathit{q}}\mathit{d\gamma }{)}^{\frac{1}{q}}\}$

$\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}\{{\left({\int }_0^1(1-\mathit{\gamma }){\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma }\right)}^{\frac{1}{p}}$

$({\int }_0^1\frac{1-\mathit{\gamma }}{{\left(A_{\gamma }\right)}^{2\mathit{q}}}\left[\begin{array}{cc}|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -\mathit{\gamma }|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-(1-\mathit{\gamma })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]\mathit{d\gamma }{)}^{\frac{1}{q}}$

$+\frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{\left({\int }_0^1\mathit{\gamma }{\left|\begin{array}{cc}(\xi +1){\beta }_{\gamma }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|}^{\mathit{p}}\mathit{d\gamma }\right)}^{\frac{1}{p}}$

(3.7) $({\int }_0^1\frac{\gamma }{{\left(A_{\gamma }\right)}^{2\mathit{q}}}\left[\begin{array}{cc}|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -\mathit{\gamma }|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-(1-\mathit{\gamma })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]\mathit{d\gamma }{)}^{\frac{1}{q}}\}.$(3.7)

Calculating the appearing integrals in (3.7), we have

${\mathrm{\Lambda }}_1={\int }_0^1\frac{1-\mathit{\gamma }}{{\left(A_{\gamma }\right)}^{2\mathit{q}}}\mathit{d\gamma }$

$={\int }_0^1\frac{1-\mathit{\gamma }}{{\left[\gamma \left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}\right)+(1-\mathit{\gamma })\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}\right)\right]}^{2\mathit{q}}}\mathit{d\gamma }$

$=[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}{\int }_0^1(1-\mathit{\gamma })(1-\mathit{\gamma }\left(1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right){)}^{-2\mathit{q}}\mathit{d\gamma }$

$=\frac{1}{2}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},1;3,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)$

${\mathrm{\Lambda }}_2={\int }_0^1\frac{\mathit{\gamma }(1-\mathit{\gamma })}{{\left[\gamma \left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}\right)+(1-\mathit{\gamma })\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}\right)\right]}^{2\mathit{q}}}\mathit{d\gamma }$

$=[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}{\int }_0^1\mathit{\gamma }(1-\mathit{\gamma })(1-\mathit{\gamma }\left(1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right){)}^{-2\mathit{q}}\mathit{d\gamma }$

$=\frac{1}{6}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},2;4,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)$

${\mathrm{\Lambda }}_3={\int }_0^1\frac{{(1-\mathit{\gamma })}^2}{{\left[\gamma \left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}\right)+(1-\mathit{\gamma })\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}\right)\right]}^{2\mathit{q}}}\mathit{d\gamma }$

$=[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}{\int }_0^1(1-\mathit{\gamma }{)}^2(1-\mathit{\gamma }\left(1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right){)}^{-2\mathit{q}}\mathit{d\gamma }$

$=\frac{1}{3}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},1;4,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)$

${\mathrm{\Lambda }}_4={\int }_0^1\frac{\gamma }{{\left[\gamma \left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}\right)+(1-\mathit{\gamma })\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}\right)\right]}^{2\mathit{q}}}\mathit{d\gamma }$

$=[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}{\int }_0^1\mathit{\gamma }(1-\mathit{\gamma }\left(1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right){)}^{-2\mathit{q}}\mathit{d\gamma }$

$=\frac{1}{2}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},2;3,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)$

${\mathrm{\Lambda }}_5={\int }_0^1\frac{{\gamma }^2}{{\left[\gamma \left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}\right)+(1-\mathit{\gamma })\left(\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}\right)\right]}^{2\mathit{q}}}\mathit{d\gamma }$

$=[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}{\int }_0^1{\gamma }^2(1-\mathit{\gamma }\left(1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right){)}^{-2\mathit{q}}\mathit{d\gamma }$

$=\frac{1}{3}{[\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}{]}^{-2\mathit{q}}}_2{F}_1\left(2\mathit{q},3;4,1-\frac{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right).$

This ends the proof.□

Remark 3.19. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2,\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ in Theorem 3.3, then we have

$|\frac{\varphi ({\mathrm{\ell }}_1)+\mathit{\varphi }({\mathrm{\ell }}_2)}{2}-\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2}{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{\int }_{{\mathrm{\ell }}_1}^{{\mathrm{\ell }}_2}\frac{\mathit{\varphi }(\mathit{\kappa })}{{\kappa }^2}\mathit{d\kappa }|\le \frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}{2}{\left[\frac{1}{2(\mathit{p}+1)}\right]}^{\frac{1}{p}}$

$\{{\left[{\mathrm{\Lambda }}_2|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q+{\mathrm{\Lambda }}_3|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q\right]}^{\frac{1}{q}}+{\left[{\mathrm{\Lambda }}_2|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q+{\mathrm{\Lambda }}_5|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q\right]}^{\frac{1}{q}}\},$

where

${\mathrm{\Lambda }}_2={\int }_0^1\frac{\mathit{\gamma }(1-\mathit{\gamma })}{{\left(\gamma {\mathrm{\ell }}_2+(1-\mathit{\gamma }){\mathrm{\ell }}_1\right)}^{2\mathit{q}}}\mathit{d\gamma }$

$=\frac{1}{{({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}^2}[\frac{{\mathrm{\ell }}_2\left({{\mathrm{\ell }}_2}^{1-2\mathit{q}}-{{\mathrm{\ell }}_1}^{1-2\mathit{q}}\right)}{(1-2\mathit{q})}-\frac{\left({{\mathrm{\ell }}_2}^{2-2\mathit{q}}-{{\mathrm{\ell }}_1}^{2-2\mathit{q}}\right)}{(2-2\mathit{q})}]$

$-\frac{1}{{({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}^3}[\frac{{{\mathrm{\ell }}_2}^2\left({{\mathrm{\ell }}_2}^{1-2\mathit{q}}-{{\mathrm{\ell }}_1}^{1-2\mathit{q}}\right)}{(1-2\mathit{q})}+\frac{\left({{\mathrm{\ell }}_2}^{3-2\mathit{q}}-{{\mathrm{\ell }}_1}^{3-2\mathit{q}}\right)}{(3-2\mathit{q})}-\frac{2{\mathrm{\ell }}_2\left({{\mathrm{\ell }}_2}^{2-2\mathit{q}}-{{\mathrm{\ell }}_1}^{2-2\mathit{q}}\right)}{(2-2\mathit{q})}]$

${\mathrm{\Lambda }}_3={\int }_0^1\frac{{(1-\mathit{\gamma })}^2}{{\left(\gamma {\mathrm{\ell }}_2+(1-\mathit{\gamma }){\mathrm{\ell }}_1\right)}^{2\mathit{q}}}\mathit{d\gamma }$

$=\frac{1}{{({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}^3}[\frac{{{\mathrm{\ell }}_2}^2\left({{\mathrm{\ell }}_2}^{1-2\mathit{q}}-{{\mathrm{\ell }}_1}^{1-2\mathit{q}}\right)}{(1-2\mathit{q})}+\frac{\left({{\mathrm{\ell }}_2}^{3-2\mathit{q}}-{{\mathrm{\ell }}_1}^{3-2\mathit{q}}\right)}{(3-2\mathit{q})}-\frac{2{\mathrm{\ell }}_2\left({{\mathrm{\ell }}_2}^{2-2\mathit{q}}-{{\mathrm{\ell }}_1}^{2-2\mathit{q}}\right)}{(2-2\mathit{q})}],$

and ${\mathrm{\Lambda }}_5$ is calculated above. This is new in literature.

4. Comparison between Hölder and improved Hölder fractional integral inequalities

Here, we provide a comparative analysis of Hölder’s integral inequality and its improved version. In Theorem 3.3, by utilizing the enhanced form of H”older’s inequality, we derive a superior lower upper bound compared to the original form in Theorem 3.2.

Example 4.1.

If one chooses $\mathit{\varphi }(\mathit{\kappa })=\frac{1}{3}{\kappa }^3,\mathit{\kappa }>0$, then by Proposition 1.1, mapping $|\mathit{\varphi }{ }^{\mathrm{\prime }}(\mathit{\kappa }){|}^q={\kappa }^{2\mathit{q}}$ for $\mathit{q}>1$ and $\mathit{\kappa }>0$ is $\mathit{\varphi }\in \mathit{HC}(\mathbb{I})$. In the instance of ${\mathrm{\ell }}_1=2,{\mathrm{\ell }}_2=3$ and $\mathit{q}=2$, let us find the right side of the inequalities in Theorem 3.2 and Theorem 3.3 with $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ and ${\omega }_1={\mathrm{\ell }}_1$, ${\omega }_2={\mathrm{\ell }}_2$ i.e. Remark 3.18 and Remark 3.19, respectively. After reducing common factor $\frac{{\mathrm{\ell }}_1{\mathrm{\ell }}_2({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}{2}$, the right side of Remark 3.18 is

$=[\frac{1}{\mathit{p}+1}{]}^{\frac{1}{p}}[{\mu }_1|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1){|}^q+{\mu }_2|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2){|}^q{]}^{\frac{1}{q}}$

$={\left[\frac{1}{3}\right]}^{\frac{1}{2}}\times {\left[\frac{7}{648}(81)+\frac{1}{54}(16)\right]}^{\frac{1}{2}}$

$\approx \mathrm{0.624846.}$

The right side of Remark 3.19, is

$=[\frac{1}{2(\mathit{p}+1)}{]}^{\frac{1}{p}}\times \{{\left[{\mathrm{\Lambda }}_2|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q+{\mathrm{\Lambda }}_3|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q\right]}^{\frac{1}{q}}$

$+{\left[{\mathrm{\Lambda }}_2|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q+{\mathrm{\Lambda }}_5|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q\right]}^{\frac{1}{q}}\}$

$={\left[\frac{1}{6}\right]}^{\frac{1}{2}}\times \{{\left[\frac{1}{216}(81)+\frac{1}{72}(16)\right]}^{\frac{1}{2}}+{\left[\frac{1}{162}(81)[phone]\frac{\mathrm{}}{\mathrm{}}\mathrm{}\right]}^{\frac{\mathrm{}}{\mathrm{}}}\}$

$\approx \mathrm{0.624815.}$

Clearly,

$0.624846>0.624815,$

which verifies that the Remark 3.19 error estimate is more accurate than the one in the Remark 3.18. We obtain the lower upper bound better than that of the original form of Hölder inequality.

Case 1: Furthermore, let us set ${\mathrm{\ell }}_1=2$, ${\mathrm{\ell }}_2=5$ and $\mathit{q}\in [1.1,50]$ as the unknown variables. The right-hand sides of Remark 3.18 and Remark 3.19 are denoted by ${\mathfrak{F}}_1(\mathit{q})$ and ${\mathfrak{F}}_2(\mathit{q})$, respectively.

Figure 5. For the case ${\mathrm{\ell }}_1=2$, ${\mathrm{\ell }}_2=5$ and $\mathit{q}\in [1.1,50]$ the diagram for example 4.1.

Plotting ${\mathfrak{F}}_1(\mathit{q})$ and ${\mathfrak{F}}_2(\mathit{q})$ in Figure 5, which illustrates the upper bound derived in Remark 3.19 is better than that of Remark 3.18.

Case 2: Now, we set $\mathit{\xi }=1$, ${\mathrm{\ell }}_1=2$, ${\mathrm{\ell }}_2=4$, $\mathit{q}=2$ and ${\lambda }_1=[2,2.1]$, ${\lambda }_2=[3.5,4]$ as the unknown. The right-hand sides of Theorem 3.2 and Theorem 3.3 are denoted by ${\mathfrak{F}}_3({\lambda }_1,{\lambda }_2)$ and ${\mathfrak{F}}_4({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$, respectively.

Figure 6. For the case ${\mathrm{\ell }}_1=2$, ${\mathrm{\ell }}_2=4$, $\mathit{q}=2$ and ${\mathrm{\ell }}_1=[2,2.1]$, ${\mathrm{\ell }}_2=[3.5,4]$ the diagram for example 4.1.

Plotting ${\mathfrak{F}}_3({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$ and ${\mathfrak{F}}_4({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$ in Figure 6, which illustrates the upper bound derived in Theorem 3.3 is better than that of Theorem 3.2.

Theorem 4.1.

If $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex function for some fixed $\mathit{q}\ge 1$ and $\mathit{\varphi }{ }^{\mathrm{\prime }}\in \mathcal{L}[{\omega }_1,{\omega }_2]$ along with assumption $\mathrm{\Omega }$ and $0<\mathit{\xi }<1$, then the following inequality for fractional integral holds:

$|I_{{\varphi }_2}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|$

(4.1) $\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{K_7}^{1-\frac{1}{q}}({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi }){\left[\begin{array}{cc}{K}_7({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_7({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_8({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q-K_9({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}},$(4.1)

where

$K_7({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })={\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }+1){\beta }_{1-\mathit{\gamma }}(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{d\gamma }$

$K_8({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })={\int }_0^1\left|\begin{array}{cc}(\mathit{\xi }-\mathrm{\aleph })\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })-& \mathit{\hspace{0.17em}}\\ (\mathit{\xi }+1){\beta }_{1-\mathit{\gamma }}(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })& \mathit{\hspace{0.17em}}\end{array}\right|\times \frac{1}{{\left(A_{\gamma }\right)}^2}\mathit{\gamma d\gamma }$

$K_9({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })=K_7({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })-K_8({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi }),$

with $\mathrm{\aleph }=0,1,2,3,\dots$ and $\mathit{\xi }=(\mathrm{\aleph },\mathrm{\aleph }+1]$

Proof.

Similarly the proof of Theorem 3.1, by using Lemma 3.2, Power mean inequality and since $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex, by using Mercer inequality, we have (4.1).□

Remark 4.1. If we take ${\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$ in Theorem 4.1, then we have the inequality proved in (Sanli et al. 2018).

Remark 4.2. If we take ${\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$ and $\mathit{\xi }=\mathrm{\aleph }+1$ in Theorem 4.1, then we have the inequality proved in [(Sanli et al. 2020), Theorem 7].

Remark 4.3. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2$, $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ in Theorem 4.1, then we have the inequality proved in (ÍşCan 2014).

Theorem 4.2.

If $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex function for some fixed $\mathit{q}\ge 1$ and $\mathit{\varphi }{ }^{\mathrm{\prime }}\in \mathcal{L}[{\omega }_1,{\omega }_2]$ along with assumption $\mathrm{\Omega }$ and $0<\mathit{\xi }<1$, then the following inequality for fractional integral holds

$I_{{\varphi }_2}(\mathfrak{g};\mathit{\xi },{\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$

((4.(4. $\le \frac{{\mathrm{\ell }}_2-{\mathrm{\ell }}_1}{{\mathrm{\ell }}_1{\mathrm{\ell }}_2(\mathit{\xi }+1)\mathit{\beta }(\mathrm{\aleph }+1,\mathit{\xi }-\mathrm{\aleph })}{K_4}^{\frac{1}{p}}(\mathit{\xi },\mathrm{\aleph })[\begin{array}{cc}{K}_4(\mathit{\xi },\mathrm{\aleph })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_4(\mathit{\xi },\mathrm{\aleph })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_5({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q-K_6({\mathrm{\ell }}_1,{\mathrm{\ell }}_2,\mathit{\xi })|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q& \mathit{\hspace{0.17em}}\end{array}{]}^{\frac{1}{q}},$((4.(4.

where $K_4(\mathit{\xi },\mathrm{\aleph }),K_5({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$ and $K_6({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)$ are the same as Theorem 3.2 and $\mathit{\xi }=(\mathrm{\aleph },\mathrm{\aleph }+1]$.

Proof.

Similarly the proof of Theorem 3.2, by using Lemma 3.2, Hölder inequality and since $|\mathit{\varphi }{ }^{\mathrm{\prime }}{|}^q$ is harmonically convex, by using Mercer inequality, we have (4.2).□

Remark 4.4. If we take ${\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$ in Theorem 4.2, then we have the inequality proved in (Sanli et al. 2018).

Remark 4.5. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2$ and $\mathit{\xi }=\mathrm{\aleph }+1$ in Theorem 3.2, then we have the inequality proved in [(Sanli et al. 2020), Theorem 10].

Remark 4.6. If we put ${\omega }_1={\mathrm{\ell }}_1,{\omega }_2={\mathrm{\ell }}_2,\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ in Theorem 4.2, then we have the inequality proved in (ÍşCan 2014).

5 Applications

5.1 Special means

Now, for numbers $0<{\mathrm{\ell }}_1<{\mathrm{\ell }}_2$, we consider special means and j-logarithmic means:

$\mathit{A}({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)=\frac{{\mathrm{\ell }}_1+{\mathrm{\ell }}_2}{2},$

$\mathit{G}({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)=({\mathrm{\ell }}_1{\mathrm{\ell }}_2{)}^{\frac{1}{2}},$

and

$L_j({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)={\left[\frac{{{\mathrm{\ell }}_2}^{(\mathit{j}+1)}-{{\mathrm{\ell }}_1}^{(\mathit{j}+1)}}{(\mathit{j}+1)({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}\right]}^{\frac{1}{j}},\mathit{j}\in \mathit{Z}\{-1,0\}.$

Proposition 5.1.

Using the same assumptions as in Theorem 3.3, if we take $\mathit{\psi }(\mathit{\omega })=\frac{{\mathit{\omega }}^{(\mathit{k}+1)}}{\mathit{k}+1}$ with $\mathit{\omega }>0$, $\mathit{k}\ge 1$ and $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$, then we get

$|\mathit{A}\left({\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right)}^{\mathit{k}+1},{\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}\right)}^{\mathit{k}+1}\right)$

$-(\mathit{k}+1)G^2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)L_{(k-1)}^{k-1}\left(\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}\right),\left(\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}\right)\right)|$

$\le \frac{(1+k)({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}{2G^2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)}\left[\frac{1}{2(\mathit{p}+1)}\right]\{{\left[\begin{array}{cc}{\mathrm{\Lambda }}_1{({\omega }_1)}^{\mathit{kq}}+{\mathrm{\Lambda }}_1{({\omega }_2)}^{\mathit{kq}}& \mathit{\hspace{0.17em}}\\ -{\mathrm{\Lambda }}_2{({\mathrm{\ell }}_1)}^{\mathit{kq}}-{\mathrm{\Lambda }}_3{({\mathrm{\ell }}_2)}^{\mathit{kq}}& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}}$

$+{\left[\begin{array}{cc}{\mathrm{\Lambda }}_4{({\omega }_1)}^{\mathit{kq}}+{\mathrm{\Lambda }}_4{({\omega }_2)}^{\mathit{kq}}& \mathit{\hspace{0.17em}}\\ -{\mathrm{\Lambda }}_5{({\mathrm{\ell }}_1)}^{\mathit{kq}}-{\mathrm{\Lambda }}_2{({\mathrm{\ell }}_2)}^{\mathit{kq}}& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}}\},$

where ${\mathrm{\Lambda }}_1-{\mathrm{\Lambda }}_5$ are same as in Theorem 3.3.

Corollary 5.1.

Under the same assumptions as in Proposition 5.1, if we take ${\omega }_1={\mathrm{\ell }}_1$ and ${\omega }_2={\mathrm{\ell }}_2$, then we get

$|\mathit{A}({{\mathrm{\ell }}_1}^{\mathit{k}+1},{{\mathrm{\ell }}_2}^{\mathit{k}+1})-G^2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)L_{(k-1)}^{k-1}({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)|$

$\le (1+\mathit{k})\frac{G^2({\mathrm{\ell }}_1,{\mathrm{\ell }}_2)({\mathrm{\ell }}_2-{\mathrm{\ell }}_1)}{2}[\frac{1}{2(\mathit{p}+1)}{]}^{\frac{1}{p}}\{[{\mathrm{\Lambda }}_3({\mathrm{\ell }}_1{)}^{\mathit{kq}}+{\mathrm{\Lambda }}_2({\mathrm{\ell }}_2{)}^{\mathit{kq}}{]}^{\frac{1}{q}}$

$+[{\mathrm{\Lambda }}_2({\mathrm{\ell }}_1{)}^{\mathit{kq}}+{\mathrm{\Lambda }}_5({\mathrm{\ell }}_2{)}^{\mathit{kq}}{]}^{\frac{1}{q}}\},$

where ${\mathrm{\Lambda }}_2-{\mathrm{\Lambda }}_5$ are same as in Remark 3.19.

5.2 Applications to Numerical Quadrature Rule

In this section, we will extend the idea by giving applications to numerical quadrature rule for harmonic convex inequality.

Theorem 5.1.

Under the assumption of Theorem 3.2 for $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$, let ${\mathbb{I}}_{\mathfrak{n}}:{\omega }_1={\kappa }_0<{\kappa }_1<{\kappa }_2\dots <{\kappa }_{\mathfrak{n}-1}<{\kappa }_{\mathfrak{n}}={\omega }_2$ is a partition of $[{\omega }_1,{\omega }_2]$, ${\mathrm{\ell }}_1,{\mathrm{\ell }}_2\in [{\kappa }_{\mathfrak{i}},{\kappa }_{\mathfrak{i}+1}]$ and ${\mathfrak{g}}_{\mathfrak{i}}=({\kappa }_{\mathfrak{i}+1}-{\kappa }_{\mathfrak{i}})$, $\mathfrak{i}=0,1,\dots ,\mathfrak{n}-1$, then we have:

${\int }_{\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_1}}}^{\frac{1}{\frac{1}{{\omega }_1}+\frac{1}{{\omega }_2}-\frac{1}{{\mathrm{\ell }}_2}}}\frac{\mathit{\varphi }(\mathit{\kappa })}{{\kappa }^2}\mathit{d\kappa }=\mathit{B}({\mathbb{I}}_{\mathfrak{n}},\mathit{\varphi })+\mathit{R}({\mathbb{I}}_{\mathfrak{n}},\mathit{\varphi }),$

where

(5.1) $\mathit{B}({\mathbb{I}}_{\mathfrak{n}},\mathit{\varphi })=\sum _{\mathfrak{i}=0}^{\mathfrak{n}-1}\frac{\mathit{\varphi }\left(\frac{1}{\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}}}+\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}+1}}-\frac{1}{{\mathit{\kappa }}_{\mathfrak{i},2}}}\right)+\mathit{\varphi }\left(\frac{1}{\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}}}+\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}+1}}-\frac{1}{{\mathit{\kappa }}_{\mathfrak{i},1}}}\right)}{2{\kappa }_{\mathfrak{i},1}{\kappa }_{\mathfrak{i},2}}{\mathfrak{g}}_{\mathfrak{i}},$(5.1)

and the remainder term satisfy the estimation:

$|\mathit{R}({\mathbb{I}}_{\mathfrak{n}},\mathit{\varphi })|$

$\le \sum _{\mathfrak{i}=0}^{\mathfrak{n}-1}\frac{{{\mathfrak{g}}_{\mathfrak{i}}}^2}{2({\kappa }_{\mathfrak{i},1}{\kappa }_{\mathfrak{i},2}{)}^2}{K_4}^{\frac{1}{p}}{\left[\begin{array}{cc}{K}_5|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_5|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_6|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q-K_7|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}},$

where $K_4-K_6$ are same as calculated in Remark 3.18.

Proof.

Applying Theorem 3.2 with $\mathit{\xi }=\mathrm{\aleph }+1$ after that $\mathit{\xi }=1$ on interval $[{\kappa }_{\mathfrak{i}},{\kappa }_{\mathfrak{i}+1}]$, $\mathfrak{i}=0,1,\dots ,\mathfrak{n}-1$, we get

$|\frac{\mathit{\varphi }\left(\frac{1}{\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}}}+\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}+1}}-\frac{1}{{\mathit{\kappa }}_{\mathfrak{i},2}}}\right)+\mathit{\varphi }\left(\frac{1}{\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}}}+\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}+1}}-\frac{1}{{\mathit{\kappa }}_{\mathfrak{i},1}}}\right)}{2{\kappa }_{\mathfrak{i},1}{\kappa }_{\mathfrak{i},2}}{\mathfrak{g}}_{\mathfrak{i}}-{\int }_{\frac{1}{\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}}}+\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}+1}}-\frac{1}{{\mathit{\kappa }}_{\mathfrak{i},1}}}}^{\frac{1}{\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}}}+\frac{1}{{\mathit{\kappa }}_{\mathfrak{i}+1}}-\frac{1}{{\mathit{\kappa }}_{\mathfrak{i},2}}}}\frac{\mathit{\varphi }(\mathit{\kappa })}{{\kappa }^2}\mathit{d\kappa }|$

$\le \frac{{{\mathfrak{g}}_{\mathfrak{i}}}^2}{2({\kappa }_{\mathfrak{i},1}{\kappa }_{\mathfrak{i},2}{)}^2}{K_4}^{\frac{1}{p}}{\left[\begin{array}{cc}{K}_5|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_1{)|}^q+K_5|\mathit{\varphi }{ }^{\mathrm{\prime }}({\omega }_2{)|}^q& \mathit{\hspace{0.17em}}\\ -K_6|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_2{)|}^q-K_7|\mathit{\varphi }{ }^{\mathrm{\prime }}({\mathrm{\ell }}_1{)|}^q& \mathit{\hspace{0.17em}}\end{array}\right]}^{\frac{1}{q}},$

for all $\mathfrak{i}=0,1,\dots ,\mathfrak{n}-1$. Summing over 0 to $\mathfrak{n}-1$ and using the triangular inequality we obtain the above estimation.□

6 Conclusion

The study dealt with the analysis of novel (H–H) Mercer type inequalities in fractional calculus for harmonic convex functions. We presented two new Conformable fractional trapezoidal type auxiliary equalities in Mercer sense. We enhanced the study of Mercer type integral inequalities using Power-mean, Hölder’s and modified Hölder integral inequalities using the novel approach. We showed by example and graph that the modified Hölder inequality offer lower upper bound than the original form. Similarly, modified power inequality offer lower upper bound than the original form. The remarkable techniques and ideas presented in this article may be developed to coordinates and fractional integral calculus. Our goal for the future is to continue and expand our research in this regard.

Acknowledgments

This study was supported via funding from the Pontifical Catholic University of Ecuador Project No. (070-UIO-2022).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the Pontificia Universidad Católica del Ecuador [Project No. 070-UIO-2022].