Hermite–Hadamard type inequality for non-convex functions employing the Caputo–Fabrizio fractional integral

ABSTRACT

Novel Hermite–Hadamard type inequalities for p-convex functions utilizing the Caputo–Fabrizio fractional integral are established. Additionally, fresh inequalities incorporating the Caputo–Fabrizio fractional integral operator are examined. Furthermore, applications to special means of the primary outcomes are presented.

KEYWORDS:

• Caputo–Fabrizio fractional integral
• convex function
• p-convex function
• Hermite–Hadamard type inequalities

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 26A51
• Secondary 26D15

1. Introduction

The theory of fractional calculus may not be considered a new subject as it has a long history; see e.g (Kilbas et al., 2003; Samko et al., 1987, 1993; Valerio et al., 2013). In the recent decade, this subject has attracted the attention of more and more researchers because of the fascinating fractional phenomenon as well as its wide application in physics, mechanics, engineering, and other areas (Failla & Zingales, 2020; Shang, 2014). For example, we can employ this framework to characterize systems, media, and domains exhibiting power law non-locality and power law memory; see (Gambo et al., 2014; Kilbas et al., 2003).

The growing interest in the field of fractional calculus has led to the development of diverse approaches for defining fractional derivatives and fractional integrals. like Baleanu et al. (2010), Caponetto (2010), Caputo (1965), Diethelm (2010), Hilfer (2003), and many more (Caputo-Fabrizio & Fabrizio, 2015; Das, 2011; Dokuyucu, Baleanu, et al., 2018; Dokuyucu, Celik, et al., 2018; Kilabas et al., 2006) in non-mathematical journals. The use of derivatives of fractional order has also spread into different fields of science besides mathematics and physics. The Hermite–Hadamard inequality is generalized by means of several fractional integral operators.

Theorem 1.1

Let $\mathit{\varphi }\to \mathbb{R}$ be a convex function defined on the interval I of real numbers and $a_1,a_2\in \mathit{I}$ with $a_1<a_2$. Then, the following inequality holds:

(1.1) $\begin{array}{l}\mathit{\varphi }\left({\displaystyle \frac{a_1+a_2}{2}}\right)\le {\displaystyle \frac{1}{a_2-a_1}}{\int }_{a_1}^{a_2}\mathit{\varphi }(\mathit{x})\mathit{dx}\le {\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}.\end{array}$(1.1)

The introduction of the Caputo–Fabrizio fractional derivative has opened up new avenues in the investigation of fractional differential equations. This derivative is formulated through the convolution of an ordinary derivative.

Another type of fractional derivatives, attributed to Hadamard and first presented in 1892 (Hadamard, 1892), distinguishes itself from the Riemann-Liouville and Caputo derivatives by incorporating logarithmic functions of arbitrary exponents within the integral kernel.

In this paper, we expanded the application of the fractional Caputo–Fabrizio derivative to the space $C_{\mathbb{R}}[0,1]$. Through this extension, we explore higher-order series-type fractional integro-differential equations. An investigation into the properties of Fractional Caputo–Fabrizio derivatives was conducted recently, as documented in reference (Atanackovic et al., 2018). The Caputo–Fabrizio fractional derivatives are discussed in the distributional setting (Atanackovic et al., 2018). A physical description of Caputo–Fabrizio fractional derivatives is presented in (Hristov, 2016). The fundamental characteristic of the Caputo–Fabrizio definition lies in its transformation of real powers into integers through Laplace transformation. This property enables the discovery of exact solutions for a variety of problems.

The present paper is summarized as follows. First, we give some preliminary material and basic definitions related to our work. Next, we develop Hermite–Hadamard type inequality via Caputo–Fabrizio fractional integral for p-convex function. Finally, we establish some fractional integral inequalities via Caputo–Fabrizio fractional integral.

2. Mathematical preliminaries

Here, we present some preliminary related to our work.

Definition 2.1.

Convexity in classical for a function$\mathit{\varphi }:\mathit{I}=[a_1,a_2]\subset \mathbb{R}\to \mathbb{R}$ is defined as

(2.1) $\begin{array}{l}\mathit{\varphi }(\mathit{\theta }{x}_1+(1-\mathit{\theta })x_2)\le \mathit{\theta \varphi }(x_1)+(1-\mathit{\theta })\mathit{\varphi }(x_2).\end{array}$(2.1)

The definition of p-convexity is given in (Zhang & Wan, 2007). Let I be a p-convex set. A function $\mathit{\varphi }:\mathit{I}\to \mathbb{R}$ is said to be p-convex function if

(2.2) $\begin{array}{l}\mathit{\varphi }(\mathit{\theta }{x}^p(1-0)y^p{)}^{\frac{1}{P}}\le \mathit{\varphi }(\mathit{x})+(1-\mathit{\theta })\mathit{\varphi }(\mathit{y}),\end{array}$(2.2)

where $\mathit{x},\mathit{y}\in \mathit{I}$ and $\mathit{\theta }\in [0,1].$

Let us recall the Caputo–Fabrizio fractional derivative.

Definition 2.2.

(See Caputo-Fabrizio & Fabrizio 2015; Abdelijawad & Baleanu 2017; Abdelijwad, 2017) Let $\mathit{\varphi }\in H^1(a_1,\mathit{a}),a_1<a_2,\mathit{\beta }\in [0,1]$. The definition of the left fractional derivatives of Caputo–Fabrizio becomes

$\left({ }_a^{CFC}{D}^{\mathit{\beta }}\mathit{\varphi }\right)(\mathit{k})={\displaystyle \frac{\mathit{A}(\mathit{\beta })}{1-\mathit{\beta }}}{\int }_a^k{\varphi }^{{\mathit{ }}^{\prime }}(\mathit{x})e^{\frac{-\mathit{\beta }(\mathit{t}-\mathit{x}{)}^{\mathit{\beta }}}{1-\mathit{\beta }}}\mathit{dx}$

with associated fractional integral given by

$\left({ }_a^{CF}{I}^{\mathit{\beta }}\mathit{\varphi }\right)(\mathit{k})={\displaystyle \frac{1-\mathit{\beta }}{\mathit{A}(\mathit{\beta })}}\mathit{\varphi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_a^k\mathit{\varphi }(\mathit{k})\mathit{dk},$

where $\mathit{A}(\mathit{\beta })>0$ is normalization function satisfying $\mathit{A}(0)=\mathit{A}(1)=1$. Regarding the right fractional derivative, we have the following

$\left({ }_a^{CFC}{D}^{\mathit{\beta }}\mathit{\varphi }\right)(\mathit{k})={\displaystyle \frac{-\mathit{A}(\mathit{\beta })}{1-\mathit{\beta }}}{\int }_k^b{\varphi }^{{\mathit{ }}^{\prime }}(\mathit{x})e^{\frac{-\mathit{\beta }(\mathit{x}-\mathit{t}{)}^{\mathit{\beta }}}{1-\mathit{\beta }}}\mathit{dx}.$

The corresponding fractional integral is expressed as follows.

$\left({ }_a^{CF}{I}_b^{\beta }\mathit{\varphi }\right)(\mathit{k})={\displaystyle \frac{1-\mathit{\beta }}{\mathit{A}(\mathit{\beta })}}\mathit{\varphi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_k^b\mathit{\varphi }(\mathit{x})\mathit{dx}.$

The generalized kernel, facilitated by Caputo–Fabrizio fractional integral operators, has been established through the work of Dragomir and Agarwal. In (Dragomir & Agarwal, 1998), the subsequent lemma is provided to aid in the derivation of enhancements for inequalities of the Hermite–Hadamard type.

Lemma 2.3.

Let $\mathit{\varphi }:\mathit{I}=[a_1,a_2]\subset \mathbb{R}\to \mathbb{R}$ be a differentiable function on I^o with $a_1<a_2$. If ${\varphi }^{{\mathit{ }}^{\prime }}\in \mathit{L}[a_1,a_2],$ then

(2.3) $\begin{array}{l}{\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}-{\displaystyle \frac{p}{a_2^p-a_1^p}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}\\ \mathit{ }={\displaystyle \frac{{\mathit{b}}^{\mathit{p}}-a_1^p}{2\mathit{p}}}{\int }_0^1{M}_p^{-1}(a_1,a_2;\mathit{\theta })(1-2\mathit{\theta }){\varphi }^{{\mathit{ }}^{\prime }}(M_{\mathit{p}}(a_1,a_2;\mathit{\theta }))\mathit{d\theta },\end{array}$(2.3)

where $M_p^{-1}(a_1,a_2;\mathit{\theta })=[\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{]}^{\frac{1}{p}-1}$.

3. An extension of the Hermite–Hadamard inequality through the utilization of the Caputo–Fabrizio fractional operator

In this section, we aim to develop Hermite–Hadamard type inequa lities for p-convex function with respect to Caputo–Fabrizio-fractional integral (Kunt & İşcan, 2017).

Theorem 3.1

Let the function $\mathit{\varphi }:[a_1,a_2]\subset \mathbb{R}\to \mathbb{R}$ be a p-convex function on $[a_1,a_2]$ and $\mathit{\varphi }\in L_1[a_1,a_2],$ where p > 0. If $\mathit{\beta }\in [0,1]$, then:

(3.1) $\begin{array}{l}\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\\ \mathit{ }\le {\displaystyle \frac{pA(\beta )}{\beta (a_2^p-a_1^p)}}[\left({ }_a^{CF}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({ }_b^{CF}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})-{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\psi }(\mathit{k})]\\ \mathit{ }\le {\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}},\end{array}$(3.1)

with $\mathit{\psi }(\mathit{x})={\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}$ holds.

Proof.

Let $\mathit{\varphi }:\mathit{I}\to \mathbb{R}$ be a p-convex function, where p > 0. We have

(3.2) $\begin{array}{l}\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\le {\displaystyle \frac{p}{a_2^p-a_1^p}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}\le {\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}.\end{array}$(3.2)

From (3.2) we have

(3.3) $\begin{array}{l}\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\le {\displaystyle \frac{p}{a_2^p-a_1^p}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx},\end{array}$(3.3)

(3.4) $\begin{array}{l}2\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\le {\displaystyle \frac{2p}{a_2^p-a_1^p}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}\\ \mathit{ }={\displaystyle \frac{2p}{a_2^p-a_1^p}}[{\int }_{a_1}^{\mathit{k}}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}+{\int }_{\mathit{k}}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}].\end{array}$(3.4)

By multiplying both side of (3.4) with ${\displaystyle \frac{\beta (a_2^p-a_1^p)}{2\mathit{pA}(\mathit{\beta })}}$ and adding ${\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\mathit{\varphi }(\mathit{k})$ we have

$\begin{array}{l}{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\mathit{\varphi }(\mathit{k})+{\displaystyle \frac{\beta (a_2^p-a_1^p)}{\mathit{pA}(\mathit{\beta })}}\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\\ \mathit{ }\le {\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\mathit{\varphi }(\mathit{k})\\ \mathit{ }+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}[{\int }_{a_1}^{\mathit{k}}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}+{\int }_{\mathit{k}}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}]\\ \mathit{ }=({\displaystyle \frac{(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\mathit{\varphi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{a_1}^{\mathit{k}}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx})\\ \mathit{ }+({\displaystyle \frac{(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\mathit{\varphi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{\mathit{k}}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}).\end{array}$

We obtain

(3.5) $\begin{array}{l}{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\mathit{\varphi }(\mathit{k})+{\displaystyle \frac{\beta (a_2^p-a_1^p)}{\mathit{pA}(\mathit{\beta })}}\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\\ \mathit{ }\le ({\displaystyle \frac{(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\mathit{\varphi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{a_1}^{\mathit{k}}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx})\\ \mathit{ }+({\displaystyle \frac{(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\mathit{\varphi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{\mathit{k}}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}).\end{array}$(3.5)

Substituting $\mathit{\psi }(\mathit{x})={\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}$ in the above equation, we obtain

$\begin{array}{l}{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\psi }(\mathit{k})+{\displaystyle \frac{\beta (a_2^p-a_1^p)}{\mathit{pA}(\mathit{\beta })}}\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\\ \mathit{ }\le ({\displaystyle \frac{(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\psi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{a_1}^{\mathit{k}}\mathit{\psi }(\mathit{x})\mathit{dx})\\ \mathit{ }+({\displaystyle \frac{(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\psi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{\mathit{k}}^{a_2}\mathit{\psi }(\mathit{x})\mathit{dx}).\end{array}$

Through the application of the Caputo–Fabrizio fractional integral definition, we obtain the following result.

(3.6) $\begin{array}{l}\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\\ \mathit{ }\le {\displaystyle \frac{pA(\beta )}{\beta (a_2^p-a_1^p)}}[\left({ }_a^{CF}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({ }_b^{CF}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})-{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}],\end{array}$(3.6)

which is the right-hand side of the theorem.

To prove the right-hand side of the theorem, we use the right-hand side of Hermite–Hadamard type inequality of p-convex function,

(3.7) $\begin{array}{l}{\displaystyle \frac{p}{a_2^p-a_1^p}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}\le {\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}.\end{array}$(3.7)

By multiplying both sides of (3.7) with ${\displaystyle \frac{\beta (a_2^p-a_1^p)}{2\mathit{pA}(\mathit{\beta })}}$ and adding ${\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\mathit{\varphi }(\mathit{k})$ we have

(3.8) $\begin{array}{l}[{\displaystyle \frac{1-\mathit{\beta }}{\mathit{A}(\mathit{\beta })}}\mathit{\psi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{a_1}^{\mathit{k}}\mathit{\psi }(\mathit{x})\mathit{dx}]\\ \mathit{ }+[{\displaystyle \frac{1-\mathit{\beta }}{\mathit{A}(\mathit{\beta })}}\mathit{\psi }(\mathit{k})+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{\mathit{k}}^{a_2}\mathit{\psi }(\mathit{x})\mathit{dx}]\\ \mathit{ }\le {\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\psi }(\mathit{k})+{\displaystyle \frac{\beta (a_2^p-a_1^p)}{2\mathit{pA}(\mathit{\beta })}}[\mathit{\varphi }(a_1)+\mathit{\varphi }(a_2)].\end{array}$(3.8)

Now, employing the definition of the Caputo–Fabrizio fractional integral, we derive the right-hand side of the theorem. Namely,

(3.9) $\begin{array}{l}{\displaystyle \frac{pA(\beta )}{\beta (a_2^p-a_1^p)}}[\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({\mathit{ }}_{a_2}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})-{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}]\\ \mathit{ }\le {\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}},\end{array}$(3.9)

which completes the proof.

Remark 3.2.

If we put p = 1 in (3.1) we will obtain Hermite–Hadamard type inequality via Caputo–Fabrizio fractional integral for classical convex function; see [Gurbuz, et al (2020) Theorem 2].

Theorem 3.3

Let $\mathit{\varphi },\mathit{\xi }:\mathit{I}\to \mathbb{R}$ be two p-convex function, where p > 0. If $\mathit{\varphi \xi }\in \mathit{L}([a_1,a_2])$, then following inequality holds

(3.10) $\begin{array}{l}{\displaystyle \frac{2pA(\beta )}{\beta (a_2^p-a_1^p)}}[\phantom{{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\xi }(\mathit{k})\mathit{\psi }(\mathit{k})}({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi \xi }(\mathit{k}))+({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi \xi }(\mathit{k}))\\ \mathit{ }-{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\xi }(\mathit{k})\mathit{\psi }(\mathit{k})]\\ \mathit{ }\le {\displaystyle \frac{2}{3}}\mathit{M}(a_1,a_2)+{\displaystyle \frac{1}{3}}\mathit{N}(a_1,a_2),\end{array}$(3.10)

with $\mathit{\psi }(\mathit{x})={\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}$and $\mathit{M}(a_1,a_2)=\mathit{\varphi }(a_1)\mathit{\xi }(a_1)+\mathit{\varphi }(a_2)\mathit{\xi }(a_2),$

$\mathit{N}(a_1,a_2)=\mathit{\varphi }(a_1)\mathit{\xi }(a_2)+\mathit{\varphi }(a_2)\mathit{\xi }(a_1)$, and $k^{{\mathit{ }}^{\prime }}\in [a_1,a_2]$, and $\mathit{A}(\mathit{\beta })>0$ is normalization function.

Proof.

Let $\mathit{\varphi },\mathit{\xi }$ are p-convex functions with p > 0, then

(3.11) $\begin{array}{l}\mathit{\varphi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\le \mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p,\\ \mathrm{\forall }{a}_1,a_2\in \mathit{I},\mathit{\theta }\in [0,1]\\ \mathit{\xi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\le \mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p,\\ \mathrm{\forall }{a}_1,a_2\in \mathit{I},\mathit{\theta }\in [0,1].\end{array}$(3.11)

Upon multiplication of the inequalities on both sides, we have the following expression.

(3.11) $\begin{array}{ll}\mathit{\varphi }(\mathit{\theta }{a}_1^p& \mathit{ }+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\mathit{\xi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\\ {\theta }^2\mathit{\varphi }(a_1)\mathit{\xi }(a_1)& \mathit{ }+(1-\mathit{\theta }{)}^2\mathit{\varphi }(a_2)\mathit{\xi }(a_2)\\ \mathit{ }+\mathit{\theta }(1-\mathit{\theta })[\mathit{\varphi }(a_1)\mathit{\xi }(a_2)+\mathit{\varphi }(a_2)\mathit{\xi }(a_1)].\end{array}$(3.11)

Integrating the (3.11) with respect to θ over $[0,1]$ and making change of variable we get

(3.12) $\begin{array}{l}{\displaystyle \frac{p}{a_1^p-a_1^p}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})\mathit{\xi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}\\ \mathit{ }\le {\displaystyle \frac{2}{3}}[\mathit{\varphi }(a_1)\mathit{\xi }(a_1)+\mathit{\varphi }(a_2)\mathit{\xi }(a_2)]\\ \mathit{ }+{\displaystyle \frac{1}{3}}[\mathit{\varphi }(a_1)\mathit{\xi }(a_2)+\mathit{\varphi }(a_2)\mathit{\xi }(a_1)].\end{array}$(3.12)

This implies in turn that,

(3.13) $\begin{array}{ll}{\displaystyle \frac{p}{a_2^p-a_1^p}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{\xi }(\mathit{x})\mathit{dx}& \mathit{ }\le {\displaystyle \frac{2}{3}}\mathit{M}(a_1,a_2)\\ \mathit{ }+{\displaystyle \frac{1}{3}}\mathit{N}(a_1,a_2).\end{array}$(3.13)

By multiplying the above equation with ${\displaystyle \frac{\beta (a_2^p-a_1^p)}{2\mathit{pA}(\mathit{\beta })}}$ and ${\displaystyle \frac{2(1-\mathit{\beta })\mathit{\varphi }(\mathit{k})\mathit{\xi }(\mathit{k})}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}$ and using the definition of Caputo–Fabrizio fractional integral we get

(3.14) $\begin{array}{l}{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}[{\int }_{a_1}^{\mathit{k}}\mathit{\psi }(\mathit{x})\mathit{\xi }(\mathit{x})\mathit{dx}+{\int }_{\mathit{k}}^{a_2}\mathit{\psi }(\mathit{x})\mathit{\xi }(\mathit{x})]\\ \mathit{ }+{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\psi }(\mathit{x})\mathit{\xi }(\mathit{x})\\ {\displaystyle \frac{\beta (a_2^p-a_1^p)}{2\mathit{pA}(\mathit{\beta })}}[{\displaystyle \frac{2}{3}}\mathit{M}(a_1,a_2)+\mathit{N}(a_1,a_2)]\\ \mathit{ }+{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\psi }(\mathit{x})\mathit{\xi }(\mathit{x}).\end{array}$(3.14)

By using the definition of Caputo–Fabrizio fractional integral we get

(3.15) $\begin{array}{l}{\displaystyle \frac{2pA(\beta )}{\beta (a_2^p-a_1^p)}}[\phantom{{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\xi }(\mathit{k})\mathit{\psi }(\mathit{k})}({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi \xi }(\mathit{k}))+({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi \xi }(\mathit{k}))\\ \mathit{ }-{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\xi }(\mathit{k})\mathit{\psi }(\mathit{k})]\\ \mathit{ }\le {\displaystyle \frac{2}{3}}\mathit{M}(a_1,a_2)+{\displaystyle \frac{1}{3}}\mathit{N}(a_1,a_2),\end{array}$(3.15)

This completes the proof.

Remark 3.4.

If we put p = 1 in Theorem 3.3, then we will obtain Theorem 3 in (Gurbuz et al., 2020).

Theorem 3.5

Let $\mathit{\varphi },\mathit{\xi }:\mathit{I}\subset \mathbb{R}\to \mathbb{R}$ be two p-convex functions. If $\mathit{\varphi \xi }\in \mathit{L}([a_1,a_2])$ with $\mathit{L}([a_1,a_2])$ being the set of integrable functions, then

(3.16) $\begin{array}{l}2\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\mathit{\xi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\\ \mathit{ }-{\displaystyle \frac{p}{a_2^p-a_1^p}}[\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi \xi }\right)(\mathit{k})+\left({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi \xi }\right)(\mathit{k})]\\ \mathit{ }+{\displaystyle \frac{(1-\beta )\psi (k)\xi )k}{\beta (a_2^p-a_1^p)}}\le {\displaystyle \frac{2}{3}}\mathit{M}(a_1,a_2)+{\displaystyle \frac{4}{3}}\mathit{N}(a_1,a_2)\end{array}$(3.16)

holds with $\mathit{\psi }(\mathit{x})={\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}.$

Proof.

Let ϕ and ξ be p-convex functions. For $\mathit{\theta }={\displaystyle \frac{1}{2}}$ we have

(3.17) $\begin{array}{l}\mathit{\varphi }{\left({\displaystyle \frac{{\mathit{a}}^{\mathit{p}}+b^{\mathit{p}}}{2}}\right)}^{\frac{1}{p}}\\ \mathit{ }\le {\displaystyle \frac{\varphi ((1-\theta )a_1^p+\mathit{\theta }{a}_2^p{)}^{\frac{1}{p}}+\mathit{\varphi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}}{2}}\end{array}$(3.17)

and

(3.18) $\begin{array}{l}\mathit{\xi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\\ \mathit{ }\le {\displaystyle \frac{\xi ((1-\theta )a_1^p+\mathit{\theta }{a}_2^p{)}^{\frac{1}{p}}+\mathit{\xi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}}{2}}.\end{array}$(3.18)

Multiplying the above inequalities (3.17) and (3.18) on both sides, we have

$\begin{array}{l}\mathit{\varphi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\mathit{\xi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\le \\ {\displaystyle \frac{1}{4}}[\mathit{\varphi }((1-\mathit{\theta })a_1^p+\mathit{\theta }{a}_2^p{)}^{\frac{1}{p}}\mathit{\xi }((1-\mathit{\theta })a_1^p+\mathit{\theta }{a}_2^p{)}^{\frac{1}{p}}\\ +\mathit{\varphi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\mathit{\xi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}]+\end{array}$

$\begin{array}{l}{\displaystyle \frac{1}{4}}[\mathit{\varphi }((1-\mathit{\theta })a_1^p+\mathit{\theta }{a}_2^p{)}^{\frac{1}{p}}\mathit{\xi }((1-\mathit{\theta })a_2^p+\mathit{\theta }{a}_1^p{)}^{\frac{1}{p}}\\ +\mathit{\varphi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\mathit{\xi }(\mathit{\theta }{a}_2^p+(1-\mathit{\theta })a_1^p{)}^{\frac{1}{p}}]\end{array}$

(3.19) $\begin{array}{l}\mathit{ }\le {\displaystyle \frac{1}{4}}[\mathit{\varphi }((1-\mathit{\theta })a_1^p+\mathit{\theta }{a}_2^p{)}^{\frac{1}{p}}\mathit{\xi }((1-\mathit{\theta })a_1^p+\mathit{\theta }{a}_2^p{)}^{\frac{1}{p}}\\ +\mathit{\varphi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\mathit{\xi }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}]\\ \mathit{ }+{\displaystyle \frac{1}{2}}[\mathit{\theta }(1-\mathit{\theta })[\mathit{\varphi }(a_1)\mathit{\xi }(a_1)+\mathit{\varphi }(a_2)\mathit{\xi }(a_2)]\\ +(1-\mathit{\theta }{)}^2\mathit{\varphi }(a_1)\mathit{\xi }(a_2)+{\theta }^2\mathit{\varphi }(a_2)\mathit{\xi }(a_1)].\end{array}$(3.19)

By integrating the preceding inequality with respect to θ over the interval $[0,1]$ and performing a change of variable, we obtain the following expression.

$\begin{array}{l}\mathit{\varphi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\mathit{\xi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\\ \mathit{ }\le {\displaystyle \frac{p}{2(a_2^p-a_1^p)}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})\mathit{\xi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}\\ \mathit{ }+{\displaystyle \frac{1}{3}}[\mathit{\varphi }(a_1)\mathit{\xi }(a_1)+\mathit{\varphi }(a_2)\mathit{\xi }(a_2)]\\ \mathit{ }+{\displaystyle \frac{2}{3}}[\mathit{\varphi }(a_1)\mathit{\xi }(a_2)+\mathit{\varphi }(a_2)\mathit{\xi }(a_1)].\end{array}$

Thus

(3.20) $\begin{array}{l}4\mathit{\varphi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\mathit{\xi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\\ \mathit{ }\le {\displaystyle \frac{p}{2(a_2^p-a_1^p)}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})\mathit{\xi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}\\ \mathit{ }+{\displaystyle \frac{4}{3}}\mathit{M}(a_1,a_2)+{\displaystyle \frac{8}{3}}\mathit{N}(a_1,a_2).\end{array}$(3.20)

By multiplying the above inequality with ${\displaystyle \frac{\beta (a_2^p-a_1^p)}{2\mathit{pA}(\mathit{\beta })}}$ and subtracting ${\displaystyle \frac{2(1-\mathit{\beta })\mathit{\varphi }(\mathit{k})\mathit{\xi }(\mathit{k})}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}$ on both sides we have,

(3.21) $\begin{array}{l}{\displaystyle \frac{2\beta (a_2^p-a_1^p)}{\mathit{pA}(\mathit{\beta })}}\mathit{\varphi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\mathit{\xi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\\ \mathit{ }-{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}[{\int }_{a_1}^{\mathit{k}}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{\xi }(\mathit{x})\mathit{dx}+{\int }_{\mathit{k}}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{\xi }(\mathit{x})\mathit{dx}]\\ \mathit{ }-{\displaystyle \frac{2(1-\mathit{\beta })\mathit{\varphi }(\mathit{k})\mathit{\xi }(\mathit{k})}{{\mathit{k}}^{1-\mathit{p}}\mathit{A}(\mathit{\beta })}}\\ \mathit{ }\le {\displaystyle \frac{\beta (a_2^p-a_1^p)}{2\mathit{pA}(\mathit{\beta })}}[{\displaystyle \frac{4}{3}}\mathit{M}(a_1,a_2)+{\displaystyle \frac{8}{3}}\mathit{N}(a_1,a_2)]\\ \mathit{ }-{\displaystyle \frac{2(1-\mathit{\beta })\mathit{\varphi }(\mathit{k})\mathit{\xi }(\mathit{k})}{{\mathit{k}}^{1-\mathit{p}}\mathit{A}(\mathit{\beta })}}.\end{array}$(3.21)

Utilizing the definition of the Caputo–Fabrizio fractional integral, we arrive at the following

(3.22) $\begin{array}{l}{\displaystyle \frac{2\beta (a_2^p-a_1^p)}{\mathit{pA}(\mathit{\beta })}}\mathit{\varphi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\mathit{\xi }{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\\ \mathit{ }-{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}[\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi \xi }\right)(\mathit{k})+\left({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi \xi }\right)(\mathit{k})]\\ \mathit{ }\le {\displaystyle \frac{\beta (a_2^p-a_1^p)}{2\mathit{pA}(\mathit{\beta })}}[{\displaystyle \frac{4}{3}}\mathit{M}(a_1,a_2)+{\displaystyle \frac{8}{3}}\mathit{N}(a_1,a_2)]\\ \mathit{ }-{\displaystyle \frac{2(1-\mathit{\beta })\mathit{\psi }(\mathit{k})\mathit{\xi }(\mathit{k})}{\mathit{A}(\mathit{\beta })}}.\end{array}$(3.22)

Multiplying the inequality with ${\displaystyle \frac{pA(\beta )}{\beta (a_2^p-a_1^p)}}$ on both sides we obtain

(3.23) $\begin{array}{l}{\displaystyle \frac{2pA(\beta )}{\beta (a_2^p-a_1^p)}}[\phantom{-{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\xi }(\mathit{k})\mathit{\psi }(\mathit{k})}({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi \xi }(\mathit{k}))+({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi \xi }(\mathit{k}))\\ -{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{A}(\mathit{\beta })}}\mathit{\xi }(\mathit{k})\mathit{\psi }(\mathit{k})]\\ \mathit{ }\le {\displaystyle \frac{2}{3}}\mathit{M}(a_1,a_2)+{\displaystyle \frac{1}{3}}\mathit{N}(a_1,a_2).\end{array}$(3.23)

This completes the proof.

Remark 3.6.

If we put p = 1 in Theorem 3.5, then we will obtain Theorem 4 in (Gurbuz et al., 2020).

4. Some new inequalities with Caputo–Fabrizio fractional operator

We will first generalize the lemma and then proceed to articulate the theorem using the lemma as a foundation.

Lemma 4.1.

Let $\mathit{\varphi }:\mathit{I}\to \mathbb{R}$ be a differentiable mapping on $I^{\mathit{o}},a_1,a_2\in \mathit{I}$ with $a_1<a_2$. If ${\varphi }^{\prime }\in L_1[a_1,a_2]$ and $\mathit{\beta }\in [0,1],$ then :

(4.1) $\begin{array}{l}{\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}{\int }_0^1(1-2\mathit{\theta })(\mathit{\theta }{a}_1^p+(1-\mathit{k\theta })a_2^p{)}^{\frac{1}{p}-1}\\ {\varphi }^{\prime }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\mathit{d\theta }-{\displaystyle \frac{2(1-\beta )\psi (k)}{\beta (a_2^p-a_1^p)}}\\ \mathit{ }={\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}-{\displaystyle \frac{pA(\beta )}{\beta (a_2^p-a_1^p)}}[\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})\\ +\left({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})],\end{array}$(4.1)

with $\mathit{\psi }(\mathit{x})={\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}$ holds, where p > 0, $\mathit{k}\in [a_1,a_2]$ and $\mathit{B}(\mathit{\beta })>0$ is a normalization function.

Proof.

Let $\mathit{\varphi }:\mathit{I}\to \mathbb{R}$ be a p-convex function with p > 0. It is easy to see that from Lemma 2.3

(4.2) $\begin{array}{l}\mathit{ }{\int }_0^1(1-2\mathit{\theta })(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}-1}{\varphi }^{\prime }(\mathit{\theta }{a}_1^p\\ \mathit{ }+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\mathit{d\theta }\\ \mathit{ }={\displaystyle \frac{p}{a_2^p-a_1^p}}\left({\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}\right)\\ \mathit{ }-{\displaystyle \frac{2{\mathit{p}}^2}{a_2^p-a_1^p}}({\int }_{a_1}^{\mathit{k}}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}+{\int }_{\mathit{k}}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}).\end{array}$(4.2)

Multiplying the above inequality with ${\displaystyle \frac{\beta (a_2^p-a_1^p{)}^2}{2p^2\mathit{A}(\mathit{\beta })}}$ and subtracting ${\displaystyle \frac{2(1-\mathit{\beta })\mathit{\varphi }(\mathit{k})}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}$,

(4.3) $\begin{array}{l}{\displaystyle \frac{\beta (a_2^p-a_1^p{)}^2}{2p^2\mathit{A}(\mathit{\beta })}}{\int }_0^1(1-2\mathit{\theta })(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}-1}\\ {\varphi }^{\prime }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\mathit{d\theta }-{\displaystyle \frac{2(1-\mathit{\beta })\mathit{\varphi }(\mathit{k})}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\\ \mathit{ }={\displaystyle \frac{a_2^p-a_1^p}{2\mathit{pA}(\mathit{\beta })}}(\mathit{\varphi }(a_1)+\mathit{\varphi }(a_2))-{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{a_1}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx}\\ \mathit{ }-{\displaystyle \frac{2(1-\mathit{\beta })\mathit{\varphi }(\mathit{k})}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\\ \mathit{ }={\displaystyle \frac{a_2^p-a_1^p}{2\mathit{pA}(\mathit{\beta })}}(\mathit{\varphi }(a_1)+\mathit{\varphi }(a_2))-({\displaystyle \frac{(1-\mathit{\beta })\mathit{\varphi }(\mathit{k})}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}\\ +{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{a_1}^{\mathit{k}}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx})\\ \mathit{ }-({\displaystyle \frac{(1-\mathit{\beta })\mathit{\varphi }(\mathit{k})}{\mathit{A}(\mathit{\beta }){\mathit{k}}^{1-\mathit{p}}}}+{\displaystyle \frac{\beta }{\mathit{A}(\mathit{\beta })}}{\int }_{\mathit{k}}^{a_2}{\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}\mathit{dx})\\ \mathit{ }={\displaystyle \frac{a_2^p-a_1^p}{2\mathit{pA}(\mathit{\beta })}}(\mathit{\varphi }(a_1)+\mathit{\varphi }(a_2))\\ \mathit{ }-[\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})].\end{array}$(4.3)

By solving the above equation we derive

(4.4) $\begin{array}{l}{\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}{\int }_0^1(1-2\mathit{\theta })(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}-1}\\ {\varphi }^{\prime }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}\mathit{d\theta }-{\displaystyle \frac{2(1-\beta )\psi (k)}{\beta (a_2^p-a_1^p)}}\\ \mathit{ }={\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}-{\displaystyle \frac{pA(\beta )}{\beta (a_2^p-a_1^p)}}\\ [\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})].\end{array}$(4.4)

The proof is then complete.

Remark 4.2.

If we put p = 1 in Theorem 4.1, then we will obtain Lemma 2 in (Gurbuz et al., 2020).

Theorem 4.3

Let $\mathit{\varphi }:\mathit{I}\to \mathit{R}$ be a different iable mapping on I^o and $\left|{\varphi }^{\prime }\right|$ be p-convex on $[a_1,a_2]$ where $a_1,a_2\in \mathit{I}$ with $a_1<a_2,\mathit{p}>0$. If ${\varphi }^{\prime }\in L_1[a_1,a_2]$ and $\mathit{\beta }\in [0,1]$, then:

(4.5) $\begin{array}{l}|{\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}+{\displaystyle \frac{2(1-\beta )\psi (k)}{\beta (a_2^p-a_1^p)}}\\ -{\displaystyle \frac{pA(\beta )}{\beta (a_2^p-a_1^p)}}[\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})]|\\ \mathit{ }\le {\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}[C_1(a_1,a_2)|{\varphi }^{\prime }(a_1)|\\ +C_2(a_1,a_2)|{\varphi }^{\prime }(a_2)|]\end{array}$(4.5)

holds, where $\begin{array}{l}{C}_1(a_1,a_2)={\int }_0^{12}{\displaystyle \frac{(1-2\theta )\theta }{[(\theta a_1^p+(1-\mathit{\theta })a_2^p){]}^{\frac{1}{p}-1}}}\mathit{dt}\\ \mathit{ }+{\int }_{12}^1{\displaystyle \frac{(2\theta -1)\theta }{[(\theta a_1^p+(1-\mathit{\theta })a_2^p){]}^{\frac{1}{p}-1}}}\mathit{d\theta },\end{array}$ and $\begin{array}{l}{C}_2(a_1,a_2)={\int }_0^{12}{\displaystyle \frac{(1-2\theta )(1-\theta )}{[(\theta a_1^p+(1-\mathit{\theta })a_2^p){]}^{\frac{1}{p}-1}}}\mathit{dt}\\ \mathit{ }+{\int }_{12}^1{\displaystyle \frac{(2\theta -1)(1-\theta )}{[(\theta a_1^p+(1-\mathit{\theta })a_2^p){]}^{\frac{1}{p}-1}}}\mathit{d\theta }.\end{array}$

Proof.

Let $\mathit{\varphi }:\mathit{I}\to \mathbb{R}$ be p-convex function. From Lemma 4.1 we have

(4.6) $\begin{array}{l}|{\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}+{\displaystyle \frac{2(1-\beta )\psi (k)}{\beta (a_2^p-a_1^p)}}\\ -{\displaystyle \frac{A(\beta )}{\beta (a_2^p-a_1^p)}}[\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})]|\\ \mathit{ }\le {\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}{\int }_0^1{\displaystyle \frac{\left(|1-2\mathit{\theta }||{\varphi }^{\prime }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}|\right)}{|(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}-1}|}}\mathit{d\theta }\\ \mathit{ }\le {\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}{\int }_0^1{\displaystyle \frac{(|1-2\mathit{\theta }|(\theta |{\varphi }^{\prime }(a_1)|+(1-\mathit{\theta })|{\varphi }^{\prime }(a_2)|))}{|(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}-1}|}}\mathit{d\theta }\\ \mathit{ }={\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}[{\int }_0^{12}{\displaystyle \frac{(1-2\theta )\theta |{\varphi }^{\prime }(a_1)|}{[(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p){]}^{\frac{1}{p}-1}}}\mathit{d\theta }\\ +{\int }_{12}^1{\displaystyle \frac{(2\theta -1)\theta |{\varphi }^{\prime }(a_1)|}{[(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p){]}^{\frac{1}{p}-1}}}\mathit{d\theta }]\\ \mathit{ }+{\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}[{\int }_0^{12}{\displaystyle \frac{(1-2\theta )(1-\theta )|{\varphi }^{\prime }(a_2)|}{[(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p){]}^{\frac{1}{p}-1}}}\mathit{d\theta }\\ +{\int }_{12}^1{\displaystyle \frac{(2\theta -1)(1-\theta )|{\varphi }^{\prime }(a_2)|}{[(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p){]}^{\frac{1}{p}-1}}}\mathit{dt\theta }]\\ \mathit{ }={\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}[C_1(a_1,a_2)|{\varphi }^{\prime }(a_1)|+C_2(a_1,a_2)|{\varphi }^{\prime }(\mathit{b})|].\end{array}$(4.6)

The proof is complete.

Remark 4.4.

If we put p = 1 in Theorem 4.3, then we will obtain Theorem 5 in (Gurbuz et al., 2020).

In (Iscan, 2019), Iscan gave a refinement of Holder integral inequality as follows.

Theorem 4.5

[Holder Iscan integral inequality (Iscan, 2019)]. Let p > 1 and $\frac{1}{p}+\frac{1}{q}=1$. If ϕ and ψ are real functions defined on interval $[a_1,a_2]$ and if ${\left|\mathit{\varphi }\right|}^{\mathit{q}}$ and ${\left|\mathit{\psi }\right|}^{\mathit{q}}$ are integrable functions on $[a_1,a_2]$ then

(4.7) $\begin{array}{l}\mathit{ }{\int }_{a_1}^{a_2}|\mathit{\varphi }(\mathit{x})\mathit{\psi }(\mathit{x})|\mathit{dx}\le {\displaystyle \frac{1}{a_2-a_1}}\\ \{{({\int }_{a_1}^{a_2}(a_2-\mathit{x}){|\mathit{\varphi }(\mathit{x})|}^{\mathit{p}}\mathit{dx})}^{\frac{1}{p}}\\ {({\int }_{a_1}^{a_2}(a_2-\mathit{x}){|\mathit{\psi }(\mathit{x})|}^{\mathit{q}}\mathit{dx})}^{\frac{1}{q}}\}+\\ {\displaystyle \frac{1}{a_2-a_1}}\{{({\int }_{a_1}^{a_2}(\mathit{x}-a_1){|\mathit{\varphi }(\mathit{x})|}^{\mathit{p}}\mathit{dx})}^{\frac{1}{p}}\\ {({\int }_{a_1}^{a_2}(\mathit{x}-a_1){|\mathit{\psi }(\mathit{x})|}^{\mathit{q}}\mathit{dx})\}}^{\frac{1}{q}}.\end{array}$(4.7)

Theorem 4.6

Let $\mathit{\varphi }:\mathit{I}\to \mathbb{R}$ be differentiable positive mappings on I^o and ${\left|{\varphi }^{\prime }\right|}^{\mathit{q}}$ is p-convex function on $[a_1,a_2]$ where $\mathit{p}>1,p^{-1}+q^{-1}=1,a_1,a_2\in \mathit{I}$ with $a_1<a_2.$ if ${\varphi }^{\prime }\in L_1[a_1,a_2]$ and $\mathit{\beta }\in [0,1],$ then the following inequality holds:

(4.8) $\begin{array}{l}|{\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}+{\displaystyle \frac{2(1-\beta )\psi (k)}{\beta (a_2^p-a_1^p)}}\\ -{\displaystyle \frac{pA(\beta )}{\beta (a_2^p-a_1^p)}}[\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})]|\\ \mathit{ }\le {\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}{({\mathit{C}}_3(a_1,a_2))}^{\frac{1}{p}}\\ {\left({\displaystyle \frac{{|{\varphi }^{\prime }(a_1)|}^{\mathit{q}}+{|{\varphi }^{\prime }(a_1)|}^{\mathit{q}}}{2}}\right)}^{\frac{1}{q}},\end{array}$(4.8)

where

$C_3(a_1,a_2)={\left({\int }_0^1{\displaystyle \frac{{|1-2\mathit{\theta }|}^{\mathit{p}}}{{|(\theta a_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}-1}|}^{\mathit{p}}}}\right)}^{\frac{1}{p}}$

and $\mathit{k}\in [a_1,a_2]$ and $\mathit{A}(\mathit{\beta })>0$ is a normalization function.

Proof.

Using the same method as that of the above theorem and Lemma 4.1, the Holder inequality and p-convexity of ${\left|{\varphi }^{\prime }\right|}^{\mathit{q}}$, we get

(4.9) $\begin{array}{l}|{\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}}+{\displaystyle \frac{2(1-\beta )\psi (k)}{\beta (a_2^p-a_1^p)}}\\ -{\displaystyle \frac{A(\beta )}{\beta (a_2^p-a_1^p)}}[\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({\mathit{ }}^{\mathit{CF}}{I}_{a_2}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})]|\\ \mathit{ }\le {\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}{\int }_0^1{\displaystyle \frac{\left(|1-2\mathit{\theta }||{\varphi }^{\prime }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}|\right)}{|(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}-1}|}}\mathit{d\theta }\\ \mathit{ }\le {\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}{\left({\int }_0^1{\displaystyle \frac{{|1-2\mathit{\theta }|}^{\mathit{p}}}{{|(\theta a_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}-1}|}^{\mathit{p}}}}\right)}^{\frac{1}{p}}\\ {\left({\int }_0^1{|{\varphi }^{\prime }(\mathit{\theta }{a}_1^p+(1-\mathit{\theta })a_2^p{)}^{\frac{1}{p}}|}^{\mathit{q}}\right)}^{\frac{1}{q}}\\ \mathit{ }={\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}{({\mathit{C}}_3(a_1,a_2))}^{\frac{1}{p}}{\left({\displaystyle \frac{{|{\varphi }^{\prime }(a_1)|}^{\mathit{q}}+{|{\varphi }^{\prime }(a_1)|}^{\mathit{q}}}{2}}\right)}^{\frac{1}{q}}.\end{array}$(4.9)

We complete the proof.

Remark 4.7.

If we insert p = 1 in Theorem 4.6, then we reproduce Theorem 6 in (Gurbuz et al., 2020).

5. Applications to special means

Means play a significant role in both applied and pure mathematics, particularly in numerical approximation, where they are utilized extensively. In literature, they are ordered in the following way

(5.1) $\begin{array}{l}\mathit{H}\le \mathit{G}\le \mathit{L}\le \mathit{I}\le \mathit{A}\end{array}$(5.1)

The arithmetic mean of $a_1,a_2$ with $a_1\ne a_2$: is defined as

(5.2) $\begin{array}{l}\mathit{A}=\mathit{A}(a_1,a_2)={\displaystyle \frac{a_1+a_2}{2}},a_1,a_2\in \mathbb{R}.\end{array}$(5.2)

The generalized logarithmic mean is defined as:

(5.3). $\begin{array}{l}\mathit{L}=L_r^r(a_1,a_2)={\displaystyle \frac{a_2^{r+1}-a_1^{r+1}}{(\mathit{r}+1)(a_2-a_1)}},\\ \mathit{r}\in \mathbb{R}\setminus \{-1\},a_1,a_2\in \mathbb{R},a_1\ne a_2.\end{array}$(5.3).

(5.4) $\begin{array}{l}{L}_{\mathit{p}}(a_1,a_2)=\left[{\left({\displaystyle \frac{a_2^{p+1}-a_1^{p+1}}{(\mathit{p}+1)(a_2-a_1)}}\right)}^{\frac{1}{p}}\right],\mathit{p}\in \mathbb{R}\setminus \{-1\}.\end{array}$(5.4)

Proposition 5.1.

Assume $a_1,a_2>0$ and $a_1<a_2$. Then

(5.5) $\begin{array}{l}{M}_{\mathit{p}}(a_1,a_2)\le L_{p-1}^{1-p}(a_1,a_2)\cdot L_p^p(a_1,a_2)\le \mathit{A}\end{array}$(5.5)

holds for $\mathit{p}\in (-\mathrm{\infty },1)\mathrm{\setminus }\{0\}$, where

$M_{\mathit{p}}(a_1,a_2)=\left[{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\right].$

Proof.

From Theorem 3.1 we have

(5.6) $\begin{array}{l}\mathit{\varphi }{\left[\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)\right]}^{\frac{1}{p}}\\ \mathit{ }\le {\displaystyle \frac{pA(\beta )}{\beta (a_2^p-a_1^p)}}\\ [\left({\mathit{ }}_{a_1}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})+\left({\mathit{ }}_{a_2}^{\mathit{CF}}{I}^{\mathit{\beta }}\mathit{\psi }\right)(\mathit{k})-{\displaystyle \frac{2(1-\mathit{\beta })}{\mathit{B}(\mathit{\beta })}}\mathit{\psi }(\mathit{k})]\\ \mathit{ }\le {\displaystyle \frac{\varphi (a_1)+\mathit{\varphi }(a_2)}{2}},\end{array}$(5.6)

where $\mathit{\psi }(\mathit{x})={\displaystyle \frac{\mathit{\varphi }(\mathit{x})}{x^{1-\mathit{p}}}}$ holds.

Setting $\mathit{\varphi }(\mathit{x})=\mathit{x},\mathit{\beta }=1$ and $\mathit{A}(\mathit{\beta })=\mathit{B}(1)=1$ in the above theorem, we obtain

(5.7) $\begin{array}{l}{\left({\displaystyle \frac{a_1^p+a_2^p}{2}}\right)}^{\frac{1}{p}}\le {\displaystyle \frac{p(a_2-a_1)}{a_2^p-a_1^p}}\left[{\displaystyle \frac{a_2^{p+1}-a_1^{p+1}}{(\mathit{p}+1)(a_2-a_1)}}\right]\\ \mathit{ }\le {\displaystyle \frac{a_1+a_2}{2}}.\end{array}$(5.7)

We have the following

$L_{\mathit{p}}(a_1,a_2)={\left({\displaystyle \frac{a_2^{p+1}-a_1^{p+1}}{(\mathit{p}+1)(a_2-a_1)}}\right)}^{\frac{1}{p}}.$

Replacing p with p −1, we have

$L_{\mathit{p}-1}(a_1,a_2)={\left({\displaystyle \frac{a_2^p-a_1^p}{\mathit{p}(a_2-a_1)}}\right)}^{\frac{1}{\mathit{p}-1}}.$

This implies that

$L_{p-1}^{p-1}(a_1,a_2)={\displaystyle \frac{a_2^p-a_1^p}{\mathit{p}(a_2-a_1)}}.$

By using these means we finally obtain

(5.8) $\begin{array}{l}{M}_{\mathit{p}}(a_1,a_2)\le L_{p-1}^{1-p}(a_1,a_2)\cdot L_p^p(a_1,a_2)\le \mathit{A}.\end{array}$(5.8)

By using Theorem 4.3 in Section 4, we have some further results regarding the special means.

Proposition 5.2.

Let $a_1,a_2\in {\mathbb{R}}^{\mathbb{+}},a_1<a_2,$ then

(5.9) $\begin{array}{l}|\mathit{A}(a_1^2,a_2^2)-\mathit{p}{L}_{p+1}^{p+1}(a_1^p,a_2^p)|\\ \mathit{ }\le {\displaystyle \frac{a_2^p-a_1^p}{\mathit{p}}}[\left|a_1\right|C_1(a_1,a_2)+\left|a_2\right|C_2(a_1,a_2)].\end{array}$(5.9)

Proof.

In Theorem 4.3, if we set $\mathit{\varphi }(\mathit{x})=x^2$ with β = 1 and $\mathit{A}(\mathit{\beta })=\mathit{A}(1)=1,$ then we obtain the result immediately.

Remark 5.3.

If we insert p = 1 in Proposition 5.2 then we will obtain Proposition 1 in (Gurbuz et al., 2020).

Proposition 5.4.

Let $a_1,a_2\in {\mathbb{R}}^{\mathbb{+}}$ and $a_1<a_2$. Then

(5.10) $\begin{array}{l}|\mathit{A}(a_1^n,a_2^n)-\mathit{p}{L}_{n-1+p}^{n-1+p}(a_1^p,a_2^p)|\le {\displaystyle \frac{a_2^p-a_1^p}{2\mathit{p}}}\\ [\left|a_1^{n-1}\right|C_1(a_1,a_2)+\left|a_2^{n-1}\right|C_2(a_1,a_2)].\end{array}$(5.10)

Proof.

In Theorem 4.3, if we set $\mathit{\varphi }(\mathit{x})=x^{\mathit{n}},\mathit{n}\in \mathbb{N}$ with β = 1 and $\mathit{B}(\mathit{\beta })=\mathit{B}(1)=1,$ then we obtain the result immediately.

Remark 5.5.

If we insert p = 1 in Proposition 5.4, then we will obtain Proposition 2 in (Gurbuz et al., 2020).

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Notes on contributors

Faisal Yasin

Faisal Yasin born in 1986, received his master’s and M. Phil degree from the University of Engineering and Technology. He earned his PhD from COMSATS University Islamabad, Lahore Campus. Currently, he is an Assistant Professor in the Department of Mathematics at the University of Lahore, Pakistan. Dr. Afzal’s research interests span several areas, including Computational Group Theory, Numerical Analysis, and Applied and Computational Mathematics. His work focuses on integrating computational techniques with mathematical theory to address complex problems in these fields, contributing to both theoretical advancements and practical applications.

Zeeshan Afzal

Zeeshan Afzal born in 1986, received his master’s degree from the University of Punjab and an M. Phil degree from GCU Lahore, Pakistan. He earned his PhD from COMSATS University Islamabad, Lahore Campus. Currently, he is an Assistant Professor in the Department of Mathematics at the University of Lahore, Pakistan. Dr. Afzal’s research interests span several areas, including Computational Neuroscience, Numerical Analysis, Applied and Computational Mathematics, and Graph Theory. His work focuses on integrating computational techniques with mathematical theory to address complex problems in these fields, contributing to both theoretical advancements and practical applications.

Muhammad Shaoib Saleem

Muhammad Shoaib Saleem is a faculty member at the University of Okara and working as chairperson in the Department of Mathematics. He has been awarded the Indigenous Scholarship and the Interim Placement for Fresh Ph.D. (IPFP) by the Higher Education Commission (HEC). Dr. Saleem is skilled in various branches of mathematics, including Functional Analysis, Mathematical Analysis, Real Analysis, and Convex Analysis.

Nazia Jahangir

Nazia Jahangir is a student in the Mathematics Department at the University of Okara in Lahore, Pakistan. She is focused on advancing her knowledge and expertise in the field of mathematics. Her primary research interests are in the application of Functional Analysis and Mathematical Analysis. Through her studies, Nazia aims to contribute to the understanding and development of these areas, exploring their various applications and implications. Her academic journey is marked by a commitment to deepening her comprehension of mathematical principles and their practical uses in solving complex problems.

Yilun Shang

Yilun Shang received his B.S. and Ph.D. degrees in Mathematics from Shanghai Jiao Tong University. He was an Associate Professor of Mathematics at Tongji University. He is currently an Associate Professor of Computer Science at Northumbria University. He is the Head of Subject in Data Science and Artificial Intelligence. His research interests include complex networks and systems, probability and analysis, algebra and combinatorics, algorithms and optimization, data science and artificial intelligence.