Interval valued Jensen’s inequalities for h-convex functions on time scales

Abstract

In the paper, we study dynamic h-convexity for interval valued functions. Some generalizations of Jensen’s inequality in interval valued analysis for h-convex functions on time scales are proved in the paper. In seek of applications of generalized Jensen’s inequality, Hermite-Hadamard type inequalities for h-convex functions on time scales are established. Further discrete analogues of newly proved results are also presented in the paper. Some numerical examples are also provided to check the validity of the results.

Keywords:

• h-convexity
• Hermite Hadamard inequality
• interval valued functions
• Jensen’s inequality
• time scales

MSC 2010:

• 28A25
• 28A78
• 28B15
• 26D15
• 34N05

1. Introduction

The theory of convex function has a long history and has been the subject of considerable research in mathematics for over a century. Convexity plays a significant part in social sciences, administration technology and theoretical development in science and current analysis. The revival of curiosity in convex functions and optimization in applied sciences and engineering can be backed down to the early 1980s. Throughout the centuries that followed, this investigation led in the emergence of convex function concept as a distinct topic of mathematical analysis.

The classical definition of convexity for a function $\psi :L\subset \mathbb{R}\to \mathbb{R}$ is $$\psi (w{a}_1+(1-w)a_2)\le w\psi (a_1)+(1-w)\psi (a_2),$$ where $a_1,a_2\in L$ and $w\in [0,1].$

By using different methodologies, the research on convexity is being expanded day after day; see Awan, Noor, Noor, and Safdar (2017). In optimization, economics, and nonlinear programming, substantially extended convexity is commonly employed. Convex functions have seen rapid generalizations and extensions in recent years; some extensions are included in Niculescu and Persson (2006), Pečarić and Tong (1992), and Zalinescu (2002). Almost all fields of mathematics and other concerns of applied sciences and engineering employ inequalities as a mechanism. Jensen inequality is widely recognized in the domain of analysis and mathematics. It is named after Johan Jensen, a Danish mathematician. The Jensen type inequalities are not only used to get the majority of classical inequalities but also to predict the effects of environmental changes (Cabrerizo & Marañón, 2022; Ruel & Ayres, 1999). Developments in the Jensen type inequalities can be seen in Andrić (2021) and Mughal, Almusawa, Haq, and Baloch (2021). Some generalizations of the Jensen type inequalities with applications can be seen in Rodić (2022). Inequalities of Jensen type for class of h-convex functions can be acquired in Lara, Merentes, and Nikodem (2016).

In Sarikaya, Saglam, and Yildirim (2008), Sarikaya et al. proved a variety of the Hermite-Hadamard inequality for functions which are h-convex. More extensions and refinements of the Hermite-Hadamard type inequalities for functions which are h-convex have been thoroughly analyzed in Dragomir (2015) and Noor, Noor, and Awan (2015). An application of Hermite Hadamard inequality for elastic torsion problem can be found in Dragomir and Keady (1998). Moreover, applications of Hermite Hadamard type inequalities to f-divergence measures as well as to some special means of real numbers and estimates for the error term of trapezoidal formula are given in Khan, Ali, and Khan (2017). Stefan Hilger established the time scales calculus in his PhD thesis 1988 (Hilger, 1990), by demonstrating ways to bring together discrete-time and continuous-time dynamical networks. Several scholars are currently interested in the calculus of time scales, which is helpful in developing theory and methodologies in biology (see, e.g. Bohner & Warth, 2007). Delta and nabla calculi are fundamental approaches to study time scales theory. Sheng, Fadag, Henderson, and Davis (2006) introduced the diamond alpha calculus by combining delta and nabla calculi. Agarwal et al. drove Jensen type inequality in Agarwal, Bohner, and Peterson (2001) with the help of delta integrals on time scales. In case of diamond-α integrals Jensen type inequality is studied in Ammi, Ferreira, and Torres (2008). In particular, for $\alpha =\frac{1}{2},$ it has the following form:

Theorem 1.1.

(Jensen’s inequality) Assume that $u_1,u_2\in \mathbb{T}$ and $u_3,u_4\in \mathbb{R}$. Imagine $\psi :[u_1,u_2]\to (u_3,u_4)$ is right dense continuous and $\Phi :(u_3,u_4)\to \mathbb{R}$ is convex. Then $$\Phi \left(\frac{\underset{u_1}{\overset{u_2}{\int }}\psi (\nu ){\diamond }_{\frac{1}{2}}\nu }{u_2-u_1}\right)\le \frac{\underset{u_1}{\overset{u_2}{\int }}\Phi (\psi (\nu )){\diamond }_{\frac{1}{2}}\nu }{u_2-u_1}.$$

Middle point Hermite-Hadamard inequality given in Dinu (2008, 5.1) is the following:

Suppose $\mathbb{T}$ is a time scale and $u,v\in \mathbb{T}.$ Assume that $\varphi :[u,v]\to \mathbb{R}$ is a continuous convex function. Then, (1) $$\varphi \left(\frac{u+v}{2}\right)\le \frac{1}{v-u}\underset{u}{\overset{v}{\int }}\varphi (\nu ){\diamond }_{\frac{1}{2}}\nu \le \frac{\varphi (u)+\varphi (v)}{2}.$$(1)

Hermite Hadamard type inequality on time scales can be seen in Mohammed, Ryoo, Kashuri, Hamed, and Abualnaja (2021).

The background of interval analysis might be linked all the way return to Archimedes’ calculation of the circle’s circumference. It was mostly neglected for quite some time due to deficiency of applications in various fields. Significant work in this field did not appear until 1950s. R. E. Moore’s acclaimed work (Moore, 1996) was the first textbook on interval analysis.

Many research publications in interval analysis are based on the presentation of an uncertain variable as an interval (Gallego-Posada & Puerta-Yepes, 2018; Markov, 1979; Yadav, Bhurjee, Karmakar, & Dikshit, 2017). Further, a thorough examination of different interval valued inequalities can be found in An, Ye, Zhao, and Liu (2019), Guo, Ye, Zhao, and Liu (2019), Liu, Ye, Zhao, and Liu (2019), and Younus, Asif, and Farhad (2015).

Recently, Jensen’s and Hermite Hadamard type inequalities for interval valued functions can be seen in Khan, Srivastava, Mohammed, Nonlaopon, and Hamed (2022).

The plan of paper is as follows: Sec. 2, consists of some basic information about interval valued arithmetics, Riemann integrable function, Riemann delta integral for interval valued functions, diamond alpha derivatives integrals and dynamic h-convexity for interval valued functions. In Sec. 3, we proved interval valued Jensen’s inequalities for h-convex functions on time scales. We apply obtained inequality for h-convex functions to produce interval valued Hermite-Hadamard inequality for h-convex functions on time scales. Further many examples are provided to check the validity of results. Finally concluding remark of the paper is given in Sec. 4.

2. Preliminaries

2.1. Interval valued arithmetics

The following arithmetics are chosen from Markov (1979) and Stefanini (2010).

Assume that $$K_C=\left\{\right[a_1,a_2]:a_1,a_2\in \mathbb{R}\}.$$

For $\left[\underset{¯}{p_1},\overline{p_1}\right],\left[\underset{¯}{q_1},\overline{q_1}\right]\in K_C,$ $$\left[\underset{¯}{p_1},\overline{p_1}\right]+\left[\underset{¯}{q_1},\overline{q_1}\right]=\left[\underset{¯}{p_1}+\underset{¯}{q_1},\overline{p_1}+\overline{q_1}\right]$$ and $$k\left[\underset{¯}{p_1},\overline{p_1}\right]=\left\{\begin{array}{c}\left[k\underset{¯}{p_1},k\overline{p_1}\right]\mathit{\text{\hspace{1em}if\hspace{1em}k}}>0,\hfill \\ \left\{0\right\}\mathit{\text{\hspace{1em}if\hspace{1em}k}}=0,\hfill \\ \left[k\overline{p_1},k\underset{¯}{p_1}\right]\mathit{\text{\hspace{1em}if\hspace{1em}k}}<0,\hfill \end{array}\right\}$$ respectively. By definition, we get $k{P}_1=P_{1}k$ $\forall$ $k\in \mathbb{R}.$ Moreover, $$\left[\underset{¯}{p_1},\overline{p_1}\right]{\ominus }_{\psi }\left[\underset{¯}{q_1},\overline{q_1}\right]=\left[\min \left\{\underset{¯}{p_1}-\underset{¯}{q_1},\overline{p_1}-\overline{q_1}\right\},\max \left\{\underset{¯}{p_1}-\underset{¯}{q_1},\overline{p_1}-\overline{q_1}\right\}\right],$$ where ${\ominus }_{\psi }$ is called gH-difference.

For $P_1=\left[\underset{¯}{p_1},\overline{p_1}\right]\in K_C,$ width of $P_1$ is referred as $\omega (P_1)=\underset{¯}{p_1}-\overline{p_1}.$ By using $\omega (.),$ we can write $$P_1{\ominus }_{\psi }{Q}_1=\left\{\begin{array}{c}\left[\underset{¯}{p_1}-\underset{¯}{q_1},\overline{p_1}-\overline{q_1}\right],\hspace{1em}\mathit{\text{if\hspace{1em}ω}}(P_1)\ge \omega (Q_1),\hfill \\ \left[\overline{p_1}-\overline{q_1},\underset{¯}{p_1}-\underset{¯}{q_1}\right],\hspace{1em}\mathit{\text{if\hspace{1em}ω}}(P_1)<\omega (Q_1).\hfill \end{array}\right\}$$

More explicitly, for $P_1,$ $Q_1,$C $\in k_C,$ we have $$P_1{\ominus }_{\psi }{Q}_1=C\iff \left\{\begin{array}{c}{P}_1=Q_1+C,\hspace{1em}\omega (P_1)\ge \omega (Q_1),\hfill \\ Q_1=P_1+(-C),\hspace{1em}\omega (P_1)<\omega (Q_1).\hfill \end{array}\right\}$$

Since $K_C$ is not totally order set (e.g. see Chalco-Cano, Flores-Franulič, & Román-Flores, 2012; Markov, 1979; Moore, 1979; Younus et al., 2015). For $P_1,Q_1\in K_C,$ such that $P_1=\left[\underset{¯}{P_1},\overline{P_1}\right],$ $Q_1=\left[\underset{¯}{Q_1},\overline{Q_1}\right],$ we say that:

1. $P_1{⪯}_{LU}{Q}_1(\mathrm{ }\text{or}\mathrm{ }{P}_1{⪯}_{LR}{Q}_1),\iff \overline{P_1}\le \overline{Q_1}\mathrm{ }\text{and}\mathrm{ }\underset{¯}{P_1}\le \underset{¯}{Q_1},P_1{\prec }_{LU}{Q}_1\mathrm{ }\text{if}\mathrm{ }{P}_1{⪯}_{LU}{Q}_1\mathrm{ }\text{and}\mathrm{ }{P}_1\ne Q_1.$

2. $P_1{⪯}_{LC}{Q}_1\iff \overline{P_1}\le \overline{Q_1}$ and $P_1{\prec }_{LC}{Q}_1\iff \overline{P_1}\le \overline{Q_1}$ if $P_1{⪯}_{LC}{Q}_1$ and $P_1\ne Q_1,$ where $m(P_1)=\frac{\underset{¯}{P_1}+\overline{P_1}}{2}.$

3. $P_1{⪯}_{UC}{Q}_1\iff \overline{P_1}\le \overline{Q_1}$ and $m(P_1)\le m(Q_1),P_1{\prec }_{UC}{Q}_1\mathrm{ }\text{if}\mathrm{ }{P}_1{⪯}_{UC}{Q}_1\mathrm{ }\text{and}\mathrm{ }{P}_1\ne Q_1.$

4. $P_1{⪯}_{CW}{Q}_1\iff m(P_1)\le m(Q_1)$ and $w(P_1)\le w(Q_1),P_1{\prec }_{CW}{Q}_1$ if $P_1{\prec }_{CW}{Q}_1$ if $P_1{⪯}_{CW}{Q}_1$ and $P_1\ne Q_1,$ where $\omega (P_1)=\underset{¯}{P_1}-\overline{P_1}.$

5. $P_1{⪯}_{LW}{Q}_1\iff \overline{P_1}\le \overline{Q_1}$ and $w(P_1)\le w(Q_1),P_1{\prec }_{LW}{Q}_1\mathrm{ }\text{if}\mathrm{ }{P}_1{⪯}_{LW}{Q}_1\mathrm{ }\text{and}\mathrm{ }{P}_1\ne Q_1.$

6. $P_1{⪯}_{UW}{Q}_1\iff \underset{¯}{P_1}\le \underset{¯}{Q_1}$ and $w(P_1)\le w(Q_1),P_1{\prec }_{UW}{Q}_1$ if $P_1{⪯}_{UW}{Q}_1$ and $P_1\ne Q_1.$

Assume that ${\mathbb{V}}^{*}=\left\{{⪯}_{LU},{⪯}_{LC},{⪯}_{UC},{⪯}_{CW},{⪯}_{LW},{⪯}_{UW}\right\}$ be the collection of these partial orders on $K_C.$ The following results include some of the properties of these partial orders.

Lemma 2.1.

Assume that ${\mathbb{V}}_1^{*}=\left\{{⪯}_{LU},{⪯}_{LC},{⪯}_{UC},{⪯}_{CW},{⪯}_{UW}\right\}.$ If $P_1{⪯}_{LW}{Q}_1,$ then $P_1{⪯}_{*}{Q}_1$ $\forall$ ${⪯}_{*}\in {\mathbb{P}}_1.$

Lemma 2.2.

Assume that ${\mathbb{V}}_2^{*}=\left\{{⪯}_{UC},{⪯}_{UW}\right\}.$ If $P_1{⪯}_{CW}{Q}_1$, then $P_1{⪯}_{*}{Q}_1$ $\forall$ ${⪯}_{*}\in {\mathbb{P}}_2.$

Lemma 2.3.

Assume that $P_1,$ $Q_1,$ C $\in K_C$. If $P_1{⪯}_{LW}{Q}_1$ and $\omega (P_1)\ge \omega (C),$ then $P_1{\ominus }_{\psi }C{⪯}_{LW}{Q}_1{\ominus }_{\psi }C.$

Lemma 2.4.

If $P_1{⪯}_{LU}{Q}_1,$ then $P_1{⪯}_{LC}{Q}_1$ and ${⪯}_{UC}{Q}_1.$

Lemma 2.5.

If $P_1{⪯}_{CW}{Q}_1,$ then $P_1{⪯}_{UC}{Q}_1$ and ${⪯}_{UW}{Q}_1.$

Lemma 2.6.

If $P_1{⪯}_{UW}{Q}_1,$ then $P_1{⪯}_{UC}{Q}_1.$

Notations: ${\mathbb{R}}_I$ denotes the collection of all intervals of $\mathbb{R}.$ ${\mathbb{R}}_I^{+}$ denotes the collection of all positive intervals of $\mathbb{R}$ and ${\mathbb{R}}_I^{-}$ denotes the collection of all negative intervals of $\mathbb{R}.$

Riemann integrable function (Zhao, An, Ye, & Liu, 2018)

Let $D^{*}$ be any finite ordered subset of $[a_1,a_2]$ obtained by division of $[a_1,a_2]$ having pattern $$D^{*}=\left\{a_1=w_0<w_1<\dots <w_{n_1}=a_2\right\}.$$

A division’s mesh represents subintervals of maximum length that make up $D^{*},$ i.e. $$\mathit{mesh}(D^{*})=\max \left\{w_j-w_{j-1}:j=1,2,\dots ,n_1\right\}.$$

Suppose that $M(\delta ,\left[a_1,a_2\right])$ is the collection of all $D^{*}\in M(\left[a_1,a_2\right])$ for which $\mathit{mesh}(D^{*})<\delta .$ In every interval $\left[w_{j-1},w_j\right],1\le j\le n_1,$ select any point ξ_j, makes the sum $$S(\varphi ,D^{*},\delta )=\sum _{j=1}^{n_1}\varphi ({\xi }_j)(w_j-w_{j-1}),$$ where $\varphi :[a_1,a_2]\to \mathbb{R}$ (or ${\mathbb{R}}_I).$ $S(\varphi ,D^{*},\delta )$ is known as Riemann sum of $\varphi$ related to $D^{*}\in M(\delta ,\left[a_1,a_2\right]).$

Definition 2.7

(Dinghas, 1956). A function $\varphi :[a_1,a_2]\to \mathbb{R}$ is known as Riemann integrable (R-integrable) on $[a_1,a_2]$ if $\exists$B $\in \mathbb{R}$ for every $ϵ>0,$ there is $\delta >0$ for which $$\left|S\left(\varphi ,M,\delta \right)-B\right|<\epsilon$$ for each Riemann sum S of $\varphi$ is in $D^{*}\in M(\delta ,\left[a_1,a_2\right])$ and independent of the option of ${\xi }_j\in \left[w_{j-1},w_j\right]$ for $1\le j\le n_1.$ Here B is known as R-integral of $\varphi$ on $[a_1,a_2]$ and is defined as $B=(R){\int }_{a_1}^{a_2}\varphi (w)dw.$ The bunch of all R-integrable functions on $[a_1,a_2]$ are indicated by $R_{([a_1,a_2])}.$

Interval valued Riemann integrable function

Definition 2.8

(Zhao et al., 2018). A function $\varphi :[a_1,a_2]\to {\mathbb{R}}_I$ is known as interval Riemann integrable (IR-integrable) on $[a_1,a_2]$ if there is B $\in {\mathbb{R}}_I,$ for all $ϵ>0,$ there exist $\delta >0$ for which $$d(S(\varphi ,M,\delta ),B)<\epsilon$$ for every Riemann sum S of $\varphi$ related to each $D^{*}\in M(\delta ,\left[a_1,a_2\right])$ and independent of the option of ${\xi }_j\in \left[w_{j-1},w_j\right]$ for $1\le j\le n_1.$ Where, B is known as the IR-integral of $\varphi$ on $[a_1,a_2]$ and is defined as $B=(IR){\int }_{a_1}^{a_2}\varphi (w)dw.$ The bunch of all IR-integrable functions on $[a_1,a_2]$ is indicated by $I{R}_{([a_1,a_2])}.$

Riemann delta integral for interval valued functions

Definition 2.9

(Zhao et al., 2018). A function $\eta :{\left[a_1,a_2\right]}_{\mathbb{T}}\to {\mathbb{R}}_I$ is known as IR $\Delta -\mathit{integrable}$ on ${\left[a_1,a_2\right]}_{\mathbb{T}}$ if $\exists$ an $B\in {\mathbb{R}}_I,\epsilon >0$ $\exists$ $\delta >0$ for which $$d\left(S(\eta ,M,\delta ),B\right)<\epsilon$$

$\forall$ $D^{*}\in M(\delta ,{[a_1,a_2]}_{\mathbb{T}}).$ In this situation, B is known as the IR $\Delta -\mathit{integral}$ of η on ${[a_1,a_2]}_{\mathbb{T}}$ and is indicated as $B=(IR){\int }_{a_1}^{a_2}\eta (w)\Delta w.$ The collection of all IR $\Delta -\mathit{integrable}$ functions on ${[a_1,a_2]}_{\mathbb{T}}$ is indicated as $I{R}_{(\Delta ,{[a_1,a_2]}_{\mathbb{T}})}.$

Theorem 2.10

(Zhao, Ye, Liu, & Torres, 2019). If $\eta \in C({[u_1,u_2]}_{\mathbb{T}},{\mathbb{R}}_I),$ then $\eta \in I{R}_{(\Delta ,{[u_1,u_2)}_{\mathbb{T}})}$ and (2) $$(IR)\underset{u_1}{\overset{u_2}{\int }}\eta (\chi )\Delta \chi =\left[\underset{u_1}{\overset{u_2}{\int }}\underset{¯}{\eta }(\chi )\Delta \chi ,\underset{u_1}{\overset{u_2}{\int }}\overline{\eta }(\chi )\Delta \chi \right].$$(2)

Theorem 2.11

(Zhao et al., 2019). Suppose that ${\zeta }_1,{\zeta }_2\in I{R}_{(\Delta ,{[u_1,u_2]}_{\mathbb{T}})}$, and α is any real number, then

1. $\alpha {\zeta }_1\in I{R}_{(\Delta ,{[u_1,u_2]}_{\mathbb{T}})}$ and $$(IR)\underset{u_1}{\overset{u_2}{\int }}\alpha {\zeta }_1(\tau )\Delta \tau =\alpha (IR)\underset{u_1}{\overset{u_2}{\int }}{\zeta }_1(\tau )\Delta \tau ;$$

2. ${\zeta }_1+{\zeta }_2\in I{R}_{(\Delta ,{[u_1,u_2]}_{\mathbb{T}})}$ and $$(IR)\underset{u_1}{\overset{u_2}{\int }}({\zeta }_1(\chi )+{\zeta }_2(\chi ))\Delta \chi =(IR)\underset{u_1}{\overset{u_2}{\int }}{\zeta }_1(\chi )\Delta \chi +(IR)\underset{u_1}{\overset{u_2}{\int }}{\zeta }_2(\chi )\Delta \chi ;$$

3. For $a_3\in {[u_1,u_2]}_{\mathbb{T}}$ and $u_1<a_3<u_2,$ $$(IR)\underset{u_1}{\overset{a_3}{\int }}{\zeta }_1(\chi )\Delta \chi +(IR)\underset{a_3}{\overset{u_2}{\int }}{\zeta }_1(\chi )\Delta \chi =(IR)\underset{u_1}{\overset{u_2}{\int }}{\zeta }_1(\chi )\Delta \chi ;$$

4. If ${\zeta }_1\subseteq g$ on ${[u_1,u_2]}_{\mathbb{T}},$ then $$(IR)\underset{u_1}{\overset{u_2}{\int }}{\zeta }_1(\tau )\Delta \tau \subseteq (IR)\underset{u_1}{\overset{u_2}{\int }}{\zeta }_2(\tau )\Delta \tau .$$

2.2. Arithmetics of diamond alpha derivative and integral

Let $\mathbb{T}$ be a time scale. If $\mathbb{T}$ has a right-scattered minimum m, then define ${\mathbb{T}}_k=\mathbb{T}-m;$ otherwise ${\mathbb{T}}_k=\mathbb{T}.$ If T has a left-scattered maximum M, then define ${\mathbb{T}}^k=\mathbb{T}-M;$ otherwise ${\mathbb{T}}^k=\mathbb{T}.$ Finally, put ${\mathbb{T}}_k^k={\mathbb{T}}^k\cap {\mathbb{T}}_k$ and for s, t $\in {\mathbb{T}}_k^k,$ denote ${\mu }_{ts}=\sigma (t)-s,$ and ${\nu }_{ts}=\rho (t)-s.$

One defines the diamond-α dynamic derivation (Sheng et al., 2006) of a function $f:\mathbb{T}\to \mathbb{R}$ at t to be the number denoted by ${\zeta }_1{}^{{\diamond }_{\alpha }}(t)$ (when it exist), with the property that, for any $\epsilon >0,$ there is a neighbourhood U of t such that for all $s\in U$ (3) $$|\alpha \left[{\zeta }_1(\sigma (t))-{\zeta }_1(s)\right]{\nu }_{ts}+(1-\alpha )\left[{\zeta }_1(\rho (t)-{\zeta }_1(s)\right]{\mu }_{ts}-{\zeta }_1^{{\diamond }_{\alpha }}(t){\mu }_{ts}{\nu }_{ts}|<\epsilon {\mu }_{ts}{\nu }_{ts}.$$(3)

A function is called diamond-α differentiable on ${\mathbb{T}}_k^k$ if ${\zeta }_1{}^{{\diamond }_{\alpha }}(\tau )$ exists for all $\tau \in {\mathbb{T}}_k^k.$

Moreover if ${\zeta }_1:\mathbb{T}\to \mathbb{R}$ is differentiable on $\mathbb{T}$ in the sense of Δ and $\nabla ,$ then ${\zeta }_1$ is diamond-α differentiable at $\tau \in {\mathbb{T}}_k^k,$ and the diamond-α derivative $$ for $0\le \alpha \le 1$ is given by (4) $${\zeta }_1{}^{{\diamond }_{\alpha }}(\tau )=\alpha {\zeta }_1{}^{\Delta }(\tau )+(1-\alpha ){\zeta }_1{}^{\nabla }(\tau ).$$(4)

We present here some properties of diamond-α derivative (Sheng et al., 2006). For that ${\zeta }_1,{\zeta }_2:\mathbb{T}\to \mathbb{R}$ be diamond-α differentiable at $\tau \in \mathbb{T}.$ Then, for $0\le \alpha \le 1$

• ${\zeta }_1+{\zeta }_2:\mathbb{T}\to \mathbb{R}$ is diamond-α differentiable at $\tau \in \mathbb{T}$ $${\left({\zeta }_1+{\zeta }_2\right)}^{{\diamond }_{\alpha }}(\tau )={\zeta }_1{}^{{\diamond }_{\alpha }}(\tau )+{\zeta }_2{}^{{\diamond }_{\alpha }}(\tau );$$

• If c $\in \mathbb{R}$ and $c{\zeta }_1:\mathbb{T}\to \mathbb{R}$ is diamond-α differentiable at $\tau \in \mathbb{T}$ and $${\left(c{\zeta }_1\right)}^{{\diamond }_{\alpha }}(\tau )=c{\zeta }_1{}^{{\diamond }_{\alpha }}(\tau );$$

• ${\zeta }_1{\zeta }_2:\mathbb{T}\to \mathbb{R}$ $${\left({\zeta }_1{\zeta }_2\right)}^{{\diamond }_{\alpha }}(\tau )={\zeta }_1{}^{{\diamond }_{\alpha }}(\tau ){\zeta }_2(\tau )+\alpha {\zeta }_1{}^{\sigma }(\tau ){\zeta }_2{}^{\Delta }(\tau )+(1-\alpha ){\zeta }_1{}^{\rho }(\tau ){\zeta }_2{}^{\nabla }(\tau ).$$

Let a, b $\to \mathbb{T}$ and ${\zeta }_1:\mathbb{T}\to \mathbb{R}.$ The diamond-α integral of ζ₁ from a to b is defined by (5) $$\underset{a}{\overset{b}{\int }}{\zeta }_1(\tau ){\diamond }_{\alpha }\tau =\alpha \underset{a}{\overset{b}{\int }}{\zeta }_1(\tau )\Delta \tau +(1-\alpha )\underset{a}{\overset{b}{\int }}{\zeta }_1(\tau )\nabla \tau ,$$(5)

For $\alpha =\frac{1}{2},$ (4)(4) $${\zeta }_1{}^{{\diamond }_{\alpha }}(\tau )=\alpha {\zeta }_1{}^{\Delta }(\tau )+(1-\alpha ){\zeta }_1{}^{\nabla }(\tau ).$$(4) and (5)(5) $$\underset{a}{\overset{b}{\int }}{\zeta }_1(\tau ){\diamond }_{\alpha }\tau =\alpha \underset{a}{\overset{b}{\int }}{\zeta }_1(\tau )\Delta \tau +(1-\alpha )\underset{a}{\overset{b}{\int }}{\zeta }_1(\tau )\nabla \tau ,$$(5) are symmetric derivative and symmetric integral (respectively) involving delta and nabla derivatives$\setminus$integrals. The present study is restricted to use of diamond-$\frac{1}{2}$ derivative or integral.

Similar to Theorem 2.10, one can define interval valued nabla integral in the form:

Theorem 2.12.

If $\eta \in C({(u_1,u_2]}_{\mathbb{T}},{\mathbb{R}}_I),$ then $\eta \in I{R}_{(\nabla ,{[u_1,u_2]}_{\mathbb{T}})}$ and (6) $$(IR)\underset{u_1}{\overset{u_2}{\int }}\eta (\chi )\nabla \chi =\left[\underset{u_1}{\overset{u_2}{\int }}\underset{¯}{\eta }(\chi )\nabla \chi ,\underset{u_1}{\overset{u_2}{\int }}\overline{\eta }(\chi )\nabla \chi \right].$$(6)

Theorem 2.10 together with Theorem 2.12 gives us the following result:

Theorem 2.13.

If $\eta \in C({[u_1,u_2]}_{\mathbb{T}},{\mathbb{R}}_I),$ then $\eta \in I{R}_{({\diamond }_{\frac{1}{2}},{[u_1,u_2]}_{\mathbb{T}})}$ and (7) $$(IR)\underset{u_1}{\overset{u_2}{\int }}\eta (\chi ){\diamond }_{\frac{1}{2}}\chi =\left[\underset{u_1}{\overset{u_2}{\int }}\underset{¯}{\eta }(\chi ){\diamond }_{\frac{1}{2}}\chi ,\underset{u_1}{\overset{u_2}{\int }}\overline{\eta }(\chi ){\diamond }_{\frac{1}{2}}\chi \right].$$(7)

Proof.

Divide (2)(2) $$(IR)\underset{u_1}{\overset{u_2}{\int }}\eta (\chi )\Delta \chi =\left[\underset{u_1}{\overset{u_2}{\int }}\underset{¯}{\eta }(\chi )\Delta \chi ,\underset{u_1}{\overset{u_2}{\int }}\overline{\eta }(\chi )\Delta \chi \right].$$(2) and (6)(6) $$(IR)\underset{u_1}{\overset{u_2}{\int }}\eta (\chi )\nabla \chi =\left[\underset{u_1}{\overset{u_2}{\int }}\underset{¯}{\eta }(\chi )\nabla \chi ,\underset{u_1}{\overset{u_2}{\int }}\overline{\eta }(\chi )\nabla \chi \right].$$(6) by 2 and add the resultants to get (7)(7) $$(IR)\underset{u_1}{\overset{u_2}{\int }}\eta (\chi ){\diamond }_{\frac{1}{2}}\chi =\left[\underset{u_1}{\overset{u_2}{\int }}\underset{¯}{\eta }(\chi ){\diamond }_{\frac{1}{2}}\chi ,\underset{u_1}{\overset{u_2}{\int }}\overline{\eta }(\chi ){\diamond }_{\frac{1}{2}}\chi \right].$$(7) . □

Remark 2.14.

Theorem 2.11 remains valid if we replace delta with nabla or diamond-$\frac{1}{2}$ integrals.

Example 2.15.

Let $\mathbb{T}=\left[-1,0\right]\cup 3^{{\mathbb{N}}_{°}},$ be a time scale and ${\mathbb{N}}_0$ is the collection of positive integers including zero. Suppose that $\eta :{[u_1,u_2]}_{\mathbb{T}}\to {\mathbb{R}}_I$ is characterized by $$\eta (\Lambda )=\left\{\begin{array}{c}\left[\Lambda ,{\Lambda }^2+1\right],\hspace{1em}\mathit{\text{if\hspace{1em}Λ}}\in 3^{{\mathbb{N}}_0},\hfill \\ \left[\Lambda ,\Lambda +1\right],\hspace{1em}\mathit{\text{if\hspace{1em}Λ}}\in \left[-1,0\right),\hfill \\ \left[1,2\right],\hspace{1em}\mathit{\text{if\hspace{1em}Λ}}=0.\hfill \end{array}\right\}$$

If ${[u_1,u_2]}_{\mathbb{T}}={\left[-1,3\right]}_{\mathbb{T}},$ then $$\begin{array}{c}(IR)\underset{-1}{\overset{3}{\int }}\varphi (\Lambda )\Delta \Lambda =\left[\underset{-1}{\overset{3}{\int }}\underset{¯}{\eta }(\Lambda )\Delta \Lambda ,\underset{-1}{\overset{3}{\int }}\overline{\eta }(\Lambda )\Delta \Lambda \right]\\ =\left[\underset{-1}{\overset{0}{\int }}\Lambda d\Lambda +\underset{0}{\overset{1}{\int }}\Delta \Lambda +\underset{1}{\overset{3}{\int }}\Lambda \Delta \Lambda ,\underset{-1}{\overset{0}{\int }}(\Lambda +1)d\Lambda +\underset{0}{\overset{1}{\int }}2\Delta \Lambda +\underset{1}{\overset{3}{\int }}({\Lambda }^2+1)\Delta \Lambda \right]\\ =\left[\frac{1}{2}{\Lambda }^2{|}_{-1}^0+1+2{\Lambda }^2{|}_1,\frac{1}{2}{\Lambda }^2+\Lambda {|}_{-1}^0+2+2\Lambda ({\Lambda }^2+1){|}_1\right]\\ =\left[2\frac{1}{2},6\frac{1}{2}\right].\\ \\ (IR)\underset{-1}{\overset{3}{\int }}\varphi (\Lambda )\nabla \Lambda =\left[\underset{-1}{\overset{3}{\int }}(\Lambda )\nabla \Lambda ,\underset{-1}{\overset{3}{\int }}\overline{\eta }(\Lambda )\nabla \Lambda \right]\\ =\left[\underset{-1}{\overset{0}{\int }}\Lambda d\Lambda +\underset{0}{\overset{1}{\int }}\Lambda \nabla \Lambda +\underset{1}{\overset{3}{\int }}\Lambda \nabla \Lambda ,\underset{-1}{\overset{0}{\int }}(\Lambda +1)d\Lambda +\underset{0}{\overset{1}{\int }}({\Lambda }^2+1)\nabla \Lambda +\underset{1}{\overset{3}{\int }}({\Lambda }^2+1)\nabla \Lambda \right]\\ =\left[\frac{1}{2}{\Lambda }^2{|}_{-1}^0+1+2{\Lambda }^2{|}_1+2{\Lambda }^2{|}_3,\left(\frac{1}{2}{\Lambda }^2+\Lambda \right){|}_{-1}^0+2+2\Lambda ({\Lambda }^2+1){|}_1+2\Lambda ({\Lambda }^2+1){|}_3\right]\\ =\left[21\frac{1}{2},66\frac{1}{2}\right].\end{array}$$

Hence $$\underset{-1}{\overset{3}{\int }}\varphi (\Lambda ){\diamond }_{\frac{1}{2}}\Lambda =\frac{1}{2}\underset{-1}{\overset{3}{\int }}\varphi (\Lambda )\Delta \Lambda +\frac{1}{2}\underset{-1}{\overset{3}{\int }}\varphi (\Lambda )\nabla \Lambda =\left[12\frac{2}{4},36\frac{2}{4}\right].$$

Remark 2.16.

Example 2.15 is given in Zhao et al. (2019) for delta integral.

We use the following notations in the next section. Let I be any interval on $\mathbb{R}$ and $I_{\mathbb{T}}=\mathbb{T}\cap I$ is interval on time scale.

Convex function on time scales

Definition 2.17.

A function $\varphi :I_{\mathbb{T}}\to \mathbb{R}$ is convex function on $I_{\mathbb{T}}$ (Dinu, 2008), if (8) $$\varphi ((1-\eta )v+\eta u)\le (1-\eta )\varphi (v)+\eta \varphi (u),$$(8) $\forall$ $v,u\in I_{\mathbb{T}}$ and all $\eta \in [0,1]$ such that $(1-\eta )v+\eta u\in I_{\mathbb{T}}.$ If the inequality $(8)$ is strict for distinct $v,u\in I_{\mathbb{T}}$ and $\eta \in (0,1),$ then function $\varphi$ is strictly convex on $I_{\mathbb{T}}.$ If $-\varphi$ is convex, then function $\varphi$ is said to be concave on $I_{\mathbb{T}}.$ $\varphi$ is affine on $I_{\mathbb{T}}$ if it is both concave and convex on $I_{\mathbb{T}}.$

Note:

Next we provide some definitions of p-function, h-convex function, h-convex function for interval valued functions on time scales which are helpful to establish our main results in Sec. 3.

P-function on time scales

Definition 2.18.

A function $\varphi :I_{\mathbb{T}}\to \mathbb{R}$ is a P-function on time scale if $\varphi$ is positive and $\forall$ $v,u\in I_{\mathbb{T}}$ and $\omega \in [0,1],$ we get $$\varphi ((1-\omega )v+\omega u)\le \varphi (v)+\varphi (u).$$

h-Convexity on time scales

Definition 2.19.

Let $h:J_{\mathbb{T}}\to \mathbb{R}$ be a positive function, $(0,1)\subseteq J_{\mathbb{T}}$ and $h(·)\ne 0.$ A function $\varphi$ is in collection $S{X}_1(h,I_{\mathbb{T}},\mathbb{R}),$ or a function $\varphi :I_{\mathbb{T}}\to \mathbb{R}$ is known as h-convex function, if $\varphi$ is positive and $\forall$ $v,u\in I_{\mathbb{T}}$ and $\eta \in (0,1)$ we get (9) $$\varphi ((1-\eta )v+\eta u)\le h(1-\eta )\varphi (v)+h(\eta )\varphi (u).$$(9)

A function $\varphi$ is called h-concave if inequality $(9)$ is reversed i.e. $\varphi \in S{V}_1(h,I_{\mathbb{T}},\mathbb{R}).$

3. Main results

In order to establish the main results, it needs to define h-convexity for interval valued functions on time scales.

3.1. h-Convexity for interval valued functions on time scales

Definition 3.1.

Suppose that $h:J_{\mathbb{T}}\to \mathbb{R}$ is a positive function such that $(0,1)\in J_{\mathbb{T}},$ and $h(·)\ne 0.$ A function Λ is in collection $S{X}_1(h,I_{\mathbb{T}},\mathbb{R})$ or a function $\left[\underset{¯}{\Lambda }(\eta ),\overline{\Lambda }(\eta )\right]:I_{\mathbb{T}}\to {\mathbb{R}}_I^{+}$ is known as h-convex function, if Λ is positive and $\forall$ $v,u\in I_{\mathbb{T}}$ and $\eta \in {[0,1]}_{\mathbb{T}},$ we get (10) $$\begin{array}{c}\left[h(1-\eta )\underset{¯}{\Lambda }(v)+h(\eta )\underset{¯}{\Lambda }(u),h(1-\eta )\overline{\Lambda }(v)+h(\eta )\overline{\Lambda }(u)\right]\\ \subseteq \left[\underset{¯}{\Lambda }\left((1-\eta )v+\eta u\right),\overline{\Lambda }\left((1-\eta )v+\eta u\right)\right].\end{array}$$(10)

A function Λ is called h-concave if set inclusion $(10)$ is reversed.

Example 3.2.

Let $I_{\mathbb{T}}=[a_1,a_2], v,u\in I_{\mathbb{T}},k<0.$ Define $h(s)=s^k, s>0$ and $$\Lambda (t)=\left\{\begin{array}{cc}[1,2^{2-k}t]& t\ne \frac{a+b}{2}\\ [2^{1-k},2^{2-k}t]& t=\frac{a_1+a_2}{2}\end{array}\right\}.$$ then $\Lambda (·)$ is not convex interval valued function but h-convex interval valued function for $t>\frac{1}{2}.$ For more simplicity, choose $k=\frac{-1}{3},s=\frac{1}{2},a_1=1,a_2=5,$ then when $(1-\eta )v+\eta u=\frac{a_1+a_2}{2}$ we get equality and when one of v or u is $\frac{a_1+a_2}{2}$ we get $\left[4.22,3.66\right]\subseteq \left[1,42.66\right].$

Notations. The set of all h-concave interval valued functions on time scales is indicated by $S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+}).$ The collection of all h-convex interval valued functions is indicated by $S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+}).$ The collection of all h-affine interval valued functions is indicated by $S{A}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+}).$

3.2. Interval valued Jensen’s inequalities for h-convex functions on time scales

Theorem 3.3.

Suppose $\Phi :I_{\mathbb{T}}\to {\mathbb{R}}_I^{+}$ is interval valued function which is h-convex on time scales and $\Phi (\varrho )=[\underset{¯}{\Phi }(\varrho ),\overline{\Phi }(\varrho )]$. Then $\Phi \in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+})$ iff $\underset{¯}{\Phi }\in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+})$ and $\overline{\Phi }\in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+}).$

Proof.

Let $\Phi \in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+})$ and ${\mu }_1,{\nu }_2\in I_{\mathbb{T}},0<\vartheta <1.$ Then $$h(1-\vartheta )\Phi ({\nu }_2)+h(\vartheta )\Phi ({\mu }_1)\subseteq \Phi ((1-\vartheta ){\nu }_2+\vartheta {\mu }_1),$$ and (11) $$\begin{array}{c}[h(1-\vartheta )\underset{¯}{\Phi }({\nu }_2)+h(\vartheta )\underset{¯}{\Phi }({\mu }_1),h(1-\vartheta )\overline{\Phi }({\nu }_2)+h(\vartheta )\overline{\Phi }({\mu }_1)]\\ \subseteq [\underset{¯}{\Phi }(1-\vartheta ){\nu }_2+\vartheta {\mu }_1),\overline{\Phi }(1-\vartheta ){\nu }_2+\vartheta {\mu }_1)].\end{array}$$(11)

It comes that $$h(1-\vartheta )\underset{¯}{\Phi }({\nu }_2)+h(\vartheta )\underset{¯}{\Phi }({\mu }_1)\ge \underset{¯}{\Phi }((1-\vartheta ){\nu }_2+\vartheta {\mu }_1),$$ and $$h(1-\vartheta )\overline{\Phi }({\nu }_2)+h(\vartheta )\overline{\Phi }({\mu }_1)\le \overline{\Phi }((1-\vartheta ){\nu }_2+\vartheta {\mu }_1).$$

This reveals that $\underset{¯}{\Phi }\in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+})$ and $\overline{\Phi }\in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+}).$ Conversely, if $\underset{¯}{\Phi }\in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+})$ and $\overline{\Phi }\in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+}),$ then from definition of P-function and set inclusion $(11),$ one obtains $\Phi \in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+}).$ □

Example 3.4.

Let $\Phi (\varrho )=[{\varrho }^2,100+\sqrt{\varrho }],I_{\mathbb{T}}=[0,1]\cup \{3,3^2\},J_{\mathbb{T}}=\{0,\frac{1}{3},\frac{2}{3},1\}$ and $h(\eta )=\eta ,$ then Theorem 3.3 is satisfied.

Corollary 3.5.

If we choose time scale to be set of real numbers in Theorem $3.3$, it becomes (Zhao et al., 2018, Theorem 3.7).

Theorem 3.6.

Suppose $\Phi :I_{\mathbb{T}}\to {\mathbb{R}}_I^{+}$ is interval valued function which is h-concave on time scales and $\Phi (\vartheta )=[\underset{¯}{\Phi }(\vartheta ),\overline{\Phi }(\vartheta )]$. Then $\Phi \in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+})$ iff $\underset{¯}{\Phi }\in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+})$ and $\overline{\Phi }\in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+}).$

Proof.

Suppose that $\Phi \in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+})$ and ${\mu }_1,{\nu }_2\in I_{\mathbb{T}},0<\vartheta <1.$ Then $$h(1-\vartheta )\Phi ({\nu }_2)+h(\vartheta )\Phi ({\mu }_1)\supseteq \Phi ((1-\vartheta ){\nu }_2+\vartheta {\mu }_1),$$ and (12) $$\begin{array}{c}[h(1-\vartheta )\underset{¯}{\Phi }({\nu }_2)+h(\vartheta )\underset{¯}{\Phi }({\mu }_1),h(1-\vartheta )\overline{\Phi }({\nu }_2)+h(\vartheta )\overline{\Phi }({\mu }_1)]\\ \supseteq [\underset{¯}{\Phi }((1-\vartheta ){\nu }_2+\vartheta {\mu }_1),\overline{\Phi }((1-\vartheta ){\nu }_2+\vartheta {\mu }_1)].\end{array}$$(12)

It comes that $$h(1-\vartheta )\underset{¯}{\Phi }({\nu }_2)+h(\vartheta )\underset{¯}{\Phi }({\mu }_1)\le \underset{¯}{\Phi }((1-\vartheta ){\nu }_2+\vartheta {\mu }_1),$$ and $$h(1-\vartheta )\overline{\Phi }({\nu }_2)+h(\vartheta )\overline{\Phi }({\mu }_1)\ge \overline{\Phi }((1-\vartheta ){\nu }_2+\vartheta {\mu }_1).$$

This reveals that $\underset{¯}{\Phi }\in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+})$ and $\overline{\Phi }\in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+}).$ Conversely, if $\underset{¯}{\Phi }\in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+})$ and $\overline{\Phi }\in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}^{+}),$ then from definition of P-function and set inclusion $(12),$ one obtains $\Phi \in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+}).$ □

Example 3.7.

Let $I_{\mathbb{T}}=[a_1,a_2],k\in (0,1).$ Define $h(s)=s^{2+k}, s>0$ and $$\Lambda (t)=\left\{\begin{array}{cc}[2^{1-k},3+t]& t\ne \frac{a_1+a_2}{2}\\ [2^k,3+t]& t=\frac{a_1+a_2}{2}\end{array}\right\}.$$ then $\Lambda (·)$ is not concave interval valued function but h-concave interval valued function.

Corollary 3.8.

If we choose time scale to be a set of real numbers in Theorem $3.6$, it becomes (Zhao et al., 2018, Theorem 3.8).

3.3. Applications to interval valued Hermite Hadamard inequalities for h-convex functions

Theorem 3.9.

Suppose that $\Phi :\mathbb{T}\to {\mathbb{R}}_I^{+}$ is a interval valued function which is h-convex and $\Phi (\eta )=[\underset{¯}{\Phi }(\eta ),\overline{\Phi }(\eta )]$ and $\Phi \in I{R}_{([\zeta ,\vartheta ])},$ $h\left(\frac{1}{2}\right)\ne 0,{\int }_0^{1}h(\eta ){\diamond }_{\frac{1}{2}}\eta \ge \frac{1}{2}$ and $h:\left[0,1\right]\to \mathbb{R}$ is a positive function. If $\Phi \in S{X}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+})$, then (13) $$\begin{array}{c}\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left(\frac{\zeta +\vartheta }{2}\right)\supseteq {\Delta }_1\supseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1\\ \supseteq {\Delta }_2\supseteq [\Phi (\zeta )+\Phi (\vartheta )]\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\underset{0}{\overset{1}{\int }}h(\eta ){\diamond }_{\frac{1}{2}}\eta ,\end{array}$$(13) where $${\Delta }_1=\frac{1}{4h\left(\frac{1}{2}\right)}\left[\Phi \left(\frac{3\zeta +\vartheta }{4}\right)+\Phi \left(\frac{\zeta +3\vartheta }{4}\right)\right],$$ and $${\Delta }_2=\left[\frac{\Phi (\zeta )+\Phi (\vartheta )}{2}+\Phi \left(\frac{\zeta +\vartheta }{2}\right)\right]\underset{0}{\overset{1}{\int }}h(\eta ){\diamond }_{\frac{1}{2}}\eta .$$

If $\Phi \in S{V}_1(h,I_{\mathbb{T}},{\mathbb{R}}_I^{+})$, then (14) $$\begin{array}{c}\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left(\frac{\zeta +\vartheta }{2}\right)\subseteq {\Delta }_1\subseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1\\ \subseteq {\Delta }_2\subseteq [\Phi (\zeta )+\Phi (\vartheta )]\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\underset{0}{\overset{1}{\int }}h(\eta ){\diamond }_{\frac{1}{2}}\eta .\end{array}$$(14)

Proof.

We split interval $[\zeta ,\vartheta ]$ into $\left[\zeta ,\frac{\zeta +\vartheta }{2}\right]$ and $\left[\frac{\zeta +\vartheta }{2},\vartheta \right].$ Then for $\left[\zeta ,\frac{\zeta +\vartheta }{2}\right],$ we have (15) $$\begin{array}{c}\Phi \left(\frac{\zeta +\frac{\zeta +\vartheta }{2}}{2}\right)=\Phi \left(\frac{(1-\eta )\frac{\zeta +\vartheta }{2}+\eta \zeta +\eta \frac{\zeta +\vartheta }{2}+(1-\eta )\zeta }{2}\right)\\ \supseteq h\left(\frac{1}{2}\right)\left[\Phi \left((1-\eta )\frac{\zeta +\vartheta }{2}+\eta \zeta \right)+\Phi \left(\eta \frac{\zeta +\vartheta }{2}+(1-\eta )\zeta \right)\right].\end{array}$$(15)

Integration of the set inclusion $(15)$ with respect to η on $[0,1],$ gives (16) $$\frac{1}{4h\left(\frac{1}{2}\right)}\Phi \left(\frac{3\zeta +\vartheta }{4}\right)\supseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\frac{\zeta +\vartheta }{2}}{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1.$$(16)

Now for $\left[\frac{\zeta +\vartheta }{2},\vartheta \right],$ we have (17) $$\begin{array}{c}\Phi \left(\frac{\frac{\zeta +\vartheta }{2}+\vartheta }{2}\right)=\Phi \left(\frac{(1-\eta )\vartheta +\eta \frac{\zeta +\vartheta }{2}+\eta \vartheta +(1-\eta )\frac{\zeta +\vartheta }{2}}{2}\right)\\ \supseteq h\left(\frac{1}{2}\right)\left[\Phi \left((1-\eta )\vartheta +\eta \frac{\zeta +\vartheta }{2}\right)+\Phi \left(\eta \vartheta +(1-\eta )\frac{\zeta +\vartheta }{2}\right)\right].\end{array}$$(17)

Integration of the set inclusion $(17)$ with respect to η on $[0,1],$ gives (18) $$\frac{1}{4h\left(\frac{1}{2}\right)}\Phi \left(\frac{\zeta +3\vartheta }{4}\right)\supseteq \frac{1}{\vartheta -\zeta }\underset{\frac{\zeta +\vartheta }{2}}{\overset{\vartheta }{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1.$$(18)

By adding inclusions (16)(16) $$\frac{1}{4h\left(\frac{1}{2}\right)}\Phi \left(\frac{3\zeta +\vartheta }{4}\right)\supseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\frac{\zeta +\vartheta }{2}}{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1.$$(16) and (18)(18) $$\frac{1}{4h\left(\frac{1}{2}\right)}\Phi \left(\frac{\zeta +3\vartheta }{4}\right)\supseteq \frac{1}{\vartheta -\zeta }\underset{\frac{\zeta +\vartheta }{2}}{\overset{\vartheta }{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1.$$(18) , one gets $$\begin{array}{c}{\Delta }_1=\frac{1}{4h\left(\frac{1}{2}\right)}\left[\Phi \left(\frac{3\zeta +\vartheta }{4}\right)+\Phi \left(\frac{\zeta +3\vartheta }{4}\right)\right]\\ \supseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1\\ =\frac{1}{2}\left[\frac{2}{\vartheta -\zeta }\underset{\zeta }{\overset{\frac{\zeta +\vartheta }{2}}{\int }}\Phi ({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1+\frac{2}{\vartheta -\zeta }\underset{\frac{\zeta +\vartheta }{2}}{\overset{\vartheta }{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1\right].\end{array}$$

By using $(1)$ and ${\int }_0^{1}h(\eta ){\diamond }_{\frac{1}{2}}\eta \ge \frac{1}{2}$ $$\begin{array}{c}{\Delta }_1\supseteq \frac{1}{2}\left[\left[\Phi (\zeta )+\Phi \left(\frac{\zeta +\vartheta }{2}\right)\right]\underset{0}{\overset{1}{\int }}h(\eta ){\diamond }_{\frac{1}{2}}\eta \right]+\frac{1}{2}\left[\left[\Phi \left(\frac{\zeta +\vartheta }{2}\right)+\Phi (\vartheta )\right]\underset{0}{\overset{1}{\int }}h(\eta ){\diamond }_{\frac{1}{2}}\eta \right]\\ =\frac{1}{2}\left[\left\{\Phi (\zeta )+\Phi (\vartheta )+2\Phi \left(\frac{\zeta +\vartheta }{2}\right)\right\}\underset{0}{\overset{1}{\int }}h(\eta ){\diamond }_{\frac{1}{2}}\eta \right]\\ =\left[\frac{\Phi (\zeta )+\Phi (\vartheta )}{2}+\Phi \left(\frac{\zeta +\vartheta }{2}\right)\right]\underset{0}{\overset{1}{\int }}h(\eta ){\diamond }_{\frac{1}{2}}\eta ={\Delta }_2.\end{array}$$

Now $$\begin{array}{c}\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left(\frac{\zeta +\vartheta }{2}\right)=\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left[\frac{1}{2}\frac{3\zeta +\vartheta }{4}+\frac{1}{2}\frac{\zeta +3\vartheta }{4}\right]\\ \supseteq \frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\left[h\left(\frac{1}{2}\right)\left\{\Phi \left(\frac{3\zeta +\vartheta }{4}\right)+\Phi \left(\frac{\zeta +3\vartheta }{4}\right)\right\}\right]\\ =\frac{1}{4\left[h\left(\frac{1}{2}\right)\right]}\left\{\Phi \left(\frac{3\zeta +\vartheta }{4}\right)+\Phi \left(\frac{\zeta +3\vartheta }{4}\right)\right\}={\Delta }_1\\ \supseteq \frac{1}{4\left[h\left(\frac{1}{2}\right)\right]}\left[h\left(\frac{1}{2}\right)\left\{\Phi (\zeta )+\Phi \left(\frac{\zeta +\vartheta }{2}\right)\right\}+h\left(\frac{1}{2}\right)\left\{\Phi \left(\frac{\zeta +\vartheta }{2}\right)+\Phi \left(\vartheta \right)\right\}\right]\\ =\frac{1}{4\left[h\left(\frac{1}{2}\right)\right]}\left[h\left(\frac{1}{2}\right)\left\{\Phi \left(\zeta \right)+\Phi \left(\vartheta \right)+2\Phi \left(\frac{\zeta +\vartheta }{2}\right)\right\}\right]\\ =\frac{1}{4}\left[\Phi \left(\zeta \right)+\Phi \left(\vartheta \right)+2\Phi \left(\frac{\zeta +\vartheta }{2}\right)\right]=\frac{1}{2}\left[\frac{\Phi \left(\zeta \right)+\Phi \left(\vartheta \right)}{2}+\Phi \left(\frac{\zeta +\vartheta }{2}\right)\right]\\ \supseteq \left[\frac{\Phi \left(\zeta \right)+\Phi \left(\vartheta \right)}{2}+\Phi \left(\frac{\zeta +\vartheta }{2}\right)\right]\underset{0}{\overset{1}{\int }}h\left(\eta \right){\diamond }_{\frac{1}{2}}\eta ={\Delta }_2\\ \supseteq \left[\frac{\Phi \left(\zeta \right)+\Phi \left(\vartheta \right)}{2}+h\left(\frac{1}{2}\right)[\Phi \left(\zeta \right)+\Phi \left(\vartheta \right)]\right]\underset{0}{\overset{1}{\int }}h\left(\eta \right){\diamond }_{\frac{1}{2}}\eta \\ =\left[\left[\Phi \left(\zeta \right)+\Phi \left(\vartheta \right)\right]\left\{\frac{1}{2}+h\left(\frac{1}{2}\right)\right\}\right]\underset{0}{\overset{1}{\int }}h\left(\eta \right){\diamond }_{\frac{1}{2}}\eta .\end{array}$$

This completes the proof. □

Corollary 3.10.

If we choose time scale to be a set of real numbers in Theorem $3.9$, it becomes (Zhao et al., 2018, Theorem 4.3).

Example 3.11.

Consider ${[\zeta ,\vartheta ]}_{\mathbb{T}}=[0,2].$ Assume that $h(\eta )=\eta ,$ $\forall$ $\eta \in {[0,1]}_{\mathbb{T}}$ and $\Phi :{[\zeta ,\vartheta ]}_{\mathbb{T}}={[0,2]}_{\mathbb{R}}\to {\mathbb{R}}_I^{+}$ be referred by $\Phi ({\varsigma }_1)=[{\varsigma }_1{}^2,10-e^{{\varsigma }_1}]$ for all ${\varsigma }_1\in [0,2].$ We have $$\begin{array}{c}\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left(\frac{\zeta +\vartheta }{2}\right)=[1,10-e],\\ {\Delta }_1=\frac{1}{2}\left[\Phi \left(\frac{1}{2}\right),\Phi \left(\frac{3}{2}\right)\right]=\left[\frac{5}{4},10-\frac{\sqrt{e}+e\sqrt{e}}{2}\right],\\ \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varsigma }_1)\Delta {\varsigma }_1=\left[\frac{4}{3},\frac{21-e^2}{2}\right],\\ {\Delta }_2=\frac{1}{2}\left(\left[2,\frac{19-e^2}{2}\right]+\left[1,10-e\right]\right)=\left[\frac{3}{2},\frac{39-2e-e^2}{4}\right],\end{array}$$ and $$\left[\Phi (\zeta )+\Phi (\vartheta )\right]\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\underset{0}{\overset{1}{\int }}h(\eta )\Delta \eta =\left[2,\frac{19-e^2}{2}\right].$$

Then one gets $$\begin{array}{c}\left[1,10-e\right]\supseteq \left[\frac{5}{4},10-\frac{\sqrt{e}+e\sqrt{e}}{2}\right]\supseteq \left[\frac{4}{3},\frac{-(e^2-21)}{2}\right]\\ \supseteq \left[\frac{3}{2},\frac{-(e^2-2e-39)}{4}\right]\supseteq \left[2,\frac{19-e^2}{2}\right].\end{array}$$

Consequently, Theorem 3.9 is verified.

Corollary 3.12.

When $\mathbb{T}={\alpha }_1\mathbb{N}$ and $\zeta ={\alpha }_1,\vartheta =(n+1){\alpha }_1,{\alpha }_1>0$ in Theorem $3.9$, then set inclusions (13)(13) $$\begin{array}{c}\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left(\frac{\zeta +\vartheta }{2}\right)\supseteq {\Delta }_1\supseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1\\ \supseteq {\Delta }_2\supseteq [\Phi (\zeta )+\Phi (\vartheta )]\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\underset{0}{\overset{1}{\int }}h(\eta ){\diamond }_{\frac{1}{2}}\eta ,\end{array}$$(13) and (14)(14) $$\begin{array}{c}\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left(\frac{\zeta +\vartheta }{2}\right)\subseteq {\Delta }_1\subseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varsigma }_1){\diamond }_{\frac{1}{2}}{\varsigma }_1\\ \subseteq {\Delta }_2\subseteq [\Phi (\zeta )+\Phi (\vartheta )]\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\underset{0}{\overset{1}{\int }}h(\eta ){\diamond }_{\frac{1}{2}}\eta .\end{array}$$(14) turn into the following inclusions respectively, (19) $$\begin{array}{c}\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left(\frac{{\alpha }_1+(n+1){\alpha }_1}{2}\right)\supseteq {\Delta }_1\supseteq \frac{1}{n}\left(\sum _{l=2}^n\Phi (l{\alpha }_1)+\frac{\Phi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)}{2}\right)\\ \supseteq {\Delta }_2\supseteq [\Phi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)]\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\frac{1}{n}\left(\sum _{l=2}^{n}h\left(\frac{l-1}{n}\right)+\frac{h(0)+h(1)}{2}\right).\end{array}$$(19) (20) $$\begin{array}{c}\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left(\frac{{\alpha }_1+(n+1){\alpha }_1}{2}\right)\subseteq {\Delta }_1\subseteq \frac{1}{n}\left(\sum _{l=2}^n\Phi (l{\alpha }_1)+\frac{\Phi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)}{2}\right)\\ \subseteq {\Delta }_2\subseteq [\Phi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)]\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\frac{1}{n}\left(\sum _{l=2}^{n}h\left(\frac{l-1}{n}\right)+\frac{h(0)+h(1)}{2}\right),\end{array}$$(20) where $${\Delta }_1=\frac{1}{4h\left(\frac{1}{2}\right)}\left[\Phi \left(\frac{3{\alpha }_1+(n+1){\alpha }_1}{4}\right)+\Phi \left(\frac{{\alpha }_1+3(n+1){\alpha }_1}{4}\right)\right],$$ and $${\Delta }_2=\left[\frac{\Phi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)}{2}+\Phi \left(\frac{{\alpha }_1+(n+1){\alpha }_1}{2}\right)\right]\frac{1}{n}\left(\sum _{l=2}^{n}h\left(\frac{l-1}{n}\right)+\frac{h(0)+h(1)}{2}\right).$$

Example 3.13.

If we choose ${\alpha }_1=1,n=4,$ h(t) = t and $\Phi (\varsigma )=[\sqrt{\varsigma },{\varsigma }^2]$ in Corollary 3.12, (19)(19) $$\begin{array}{c}\frac{1}{4{\left[h\left(\frac{1}{2}\right)\right]}^2}\Phi \left(\frac{{\alpha }_1+(n+1){\alpha }_1}{2}\right)\supseteq {\Delta }_1\supseteq \frac{1}{n}\left(\sum _{l=2}^n\Phi (l{\alpha }_1)+\frac{\Phi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)}{2}\right)\\ \supseteq {\Delta }_2\supseteq [\Phi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)]\left[\frac{1}{2}+h\left(\frac{1}{2}\right)\right]\frac{1}{n}\left(\sum _{l=2}^{n}h\left(\frac{l-1}{n}\right)+\frac{h(0)+h(1)}{2}\right).\end{array}$$(19) becomes (21) $$[1.732,9]\subset [1.7071,10]\subset [1.69,10.5]\subset [1.675,11]\subset [1.618,13].$$(21)

Theorem 3.14.

Let $\Phi ,\Psi :I_{\mathbb{T}}={[\zeta ,\vartheta ]}_{\mathbb{T}}\to {\mathbb{R}}_I^{+}$ be two interval valued functions which are h-convex with $\Phi ({\varrho }_1)=[\underset{¯}{\Phi }({\varrho }_1),\overline{\Phi }({\varrho }_1)],\Psi ({\varrho }_1)=[\underset{¯}{\Psi }({\varrho }_1),\overline{\Psi }({\varrho }_1)]$ and $\Phi \Psi \in I{R}_{({[\zeta ,\vartheta ]}_{\mathbb{T}})}$ and $h_1,h_2:{\left[0,1\right]}_{\mathbb{T}}\to \mathbb{R}$ are positive right dense continuous functions. If $\Phi \in S{X}_1(h_1,I_{\mathbb{T}},{\mathbb{R}}_I^{+}),\Psi \in S{X}_1(h_2,I_{\mathbb{T}},{\mathbb{R}}_I^{+})$, then (22) $$\frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varrho }_1)\Psi ({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1\supseteq \phi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta +\chi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(1-\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta ,$$(22) where $\phi (\zeta ,\vartheta )=\Phi (\zeta )\Psi (\zeta )+\Phi (\vartheta )\Psi (\vartheta )$ and $\chi (\zeta ,\vartheta )=\Phi (\zeta )\Psi (\vartheta )+\Phi (\vartheta )\Psi (\zeta ).$

Proof.

Since $$\Phi \in S{X}_1(h_1,I_{\mathbb{T}},{\mathbb{R}}_I^{+})\Rightarrow \left[\underset{¯}{\Phi },\overline{\Phi }\right]\in S{X}_1(h_1,I_{\mathbb{T}},{\mathbb{R}}_I^{+})$$ and $$\Psi \in S{X}_1(h_2,I_{\mathbb{T}},{\mathbb{R}}_I^{+})\Rightarrow \left[\underset{¯}{\Psi },\overline{\Psi }\right]\in S{X}_1(h_2,I_{\mathbb{T}},{\mathbb{R}}_I^{+}).$$

Therefore $\forall$ $\zeta ,\vartheta \in I_{\mathbb{T}},\eta \in {(0,1)}_{\mathbb{T}},$ one has that $$\Phi ((1-\eta )\vartheta +\eta \zeta )\supseteq h_1(1-\eta )\Phi (\vartheta )+h_1(\eta )\Phi (\zeta )$$ and $$\Psi ((1-\eta )\vartheta +\eta \zeta )\supseteq h_2(1-\eta )\Psi (\vartheta )+h_2(\eta )\Psi (\zeta ).$$

Which can be written as $$\begin{array}{c}[\underset{¯}{\Phi }((1-\eta )\vartheta +\eta \zeta ),\overline{\Phi }((1-\eta )\vartheta +\eta \zeta )]\\ \supseteq [h_1(1-\eta )\underset{¯}{\Phi }(\vartheta )+h_1(\eta )\underset{¯}{\Phi }(\zeta ),h_1(1-\eta )\overline{\Phi }(\vartheta )+h_1(\eta )\overline{\Phi }(\zeta )]\end{array}$$ and $$\begin{array}{c}[\underset{¯}{\Psi }((1-\eta )\vartheta +\eta \zeta ),\overline{\Psi }((1-\eta )\vartheta +\eta \zeta ]\\ \supseteq [h_2(1-\eta )\underset{¯}{\Psi }(\vartheta )+h_2(\eta )\underset{¯}{\Psi }(\zeta ),h_2(1-\eta )\overline{\Psi }(\vartheta )+h_2(\eta )\overline{\Psi }(\zeta )].\end{array}$$

Since $\Phi$ and $\Psi$ are non-negative, therefore $$\begin{array}{c}[\underset{¯}{\Phi }((1-\eta )\vartheta +\eta \zeta ),\overline{\Phi }(1-\eta )\vartheta +\eta \zeta )\left]\right[\underset{¯}{\Psi }((1-\eta )\vartheta +\eta \zeta ),\overline{\Psi }((1-\eta )\vartheta +\eta \zeta )]\\ \supseteq [h_2(\eta )h_1(\eta )\underset{¯}{\Phi }(\zeta )\underset{¯}{\Psi }(\zeta ),h_2(\eta )h_1(\eta )\overline{\Phi }(\zeta )\overline{\Psi }(\zeta )]\\ +[h_2(1-\eta )h_1(\eta )\underset{¯}{\Phi }(\zeta )\underset{¯}{\Psi }(\vartheta ),h_2(1-\eta )h_1(\eta )\overline{\Phi }(\zeta )\overline{\Psi }(\vartheta )]\\ +[h_1(1-\eta )h_2(\eta )\underset{¯}{\Phi }(\vartheta )\underset{¯}{\Psi }(\zeta ),h_1(1-\eta )h_2(\eta )\overline{\Phi }(\vartheta )\overline{\Psi }(\zeta )]\\ +[h_2(1-\eta )h_1(1-\eta )\underset{¯}{\Phi }(\vartheta )\underset{¯}{\Psi }(\vartheta ),h_1(1-\eta )h_2(1-\eta )\overline{\Phi }(\vartheta )\overline{\Psi }(\vartheta )].\end{array}$$ Integrating over ${[0,1]}_{\mathbb{T}},$ we get $$\begin{array}{c}\underset{0}{\overset{1}{\int }}[\underset{¯}{\Phi }((1-\eta )\vartheta +\eta \zeta ),\overline{\Phi }((1-\eta )\vartheta +\eta \zeta )][\underset{¯}{\Psi }((1-\eta )\vartheta +\eta \zeta ),\overline{\Psi }((1-\eta )\vartheta +\eta \zeta )]{\diamond }_{\frac{1}{2}}\eta \\ \supseteq [\underset{¯}{\Phi }(\zeta )\underset{¯}{\Psi }(\zeta )\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(\eta ){\diamond }_{\frac{1}{2}}\eta ,\overline{\Phi }(\zeta )\overline{\Psi }(\zeta )\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(\eta ){\diamond }_{\frac{1}{2}}\eta ]\\ +[\underset{¯}{\Phi }(\zeta )\underset{¯}{\Psi }(\vartheta )\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta ,\overline{\Phi }(\zeta )\overline{\Psi }(\vartheta )\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta ]\end{array}$$ $$\begin{array}{c}+[\underset{¯}{\Phi }(\vartheta )\underset{¯}{\Psi }(\zeta )\underset{0}{\overset{1}{\int }}{h}_2(\eta )h_1(1-\eta ){\diamond }_{\frac{1}{2}}\eta ,\overline{\Phi }(\vartheta )\overline{\Psi }(\zeta )\underset{0}{\overset{1}{\int }}{h}_2(\eta )h_1(1-\eta ){\diamond }_{\frac{1}{2}}\eta ]\\ +[\underset{¯}{\Phi }(\vartheta )\underset{¯}{\Psi }(\vartheta )\underset{0}{\overset{1}{\int }}{h}_1(1-\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta ,\overline{\Phi }(\vartheta )\overline{\Psi }(\vartheta )\underset{0}{\overset{1}{\int }}{h}_1(1-\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta ].\\ =[\underset{¯}{\Phi }(\zeta )\underset{¯}{\Psi }(\zeta ),\overline{\Phi }(\zeta )\overline{\Psi }(\zeta )]\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(\eta ){\diamond }_{\frac{1}{2}}\eta +[\underset{¯}{\Phi }(\vartheta )\underset{¯}{\Psi }(\vartheta ),\overline{\Phi }(\vartheta )\overline{\Psi }(\vartheta )]\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta \\ +[\underset{¯}{\Phi }(\vartheta )\underset{¯}{\Psi }(\zeta ),\overline{\Phi }(\vartheta )\overline{\Psi }(\zeta )]\underset{0}{\overset{1}{\int }}{h}_2(\eta )h_1(1-\eta ){\diamond }_{\frac{1}{2}}\eta +[\underset{¯}{\Phi }(\vartheta )\underset{¯}{\Psi }(\vartheta ),\overline{\Phi }(\vartheta )\overline{\Psi }(\vartheta )]\underset{0}{\overset{1}{\int }}{h}_1(1-\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta .\\ =[\underset{¯}{\Phi }(\zeta )\underset{¯}{\Psi }(\zeta )+\underset{¯}{\Phi }(\vartheta )\underset{¯}{\Psi }(\vartheta ),\overline{\Phi }(\zeta )\overline{\Psi }(\zeta )+\overline{\Phi }(\vartheta )\overline{\Psi }(\vartheta )]\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(\eta ){\diamond }_{\frac{1}{2}}\eta \\ +[\underset{¯}{\Phi }(\zeta )\underset{¯}{\Psi }(\vartheta )+\underset{¯}{\Phi }(\vartheta )\underset{¯}{\Psi }(\zeta ),\overline{\Phi }(\zeta )\overline{\Psi }(\vartheta )+\overline{\Phi }(\vartheta )\overline{\Psi }(\zeta )]\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta .\\ =[\Phi (\zeta )\Psi (\zeta )+\Phi (\vartheta )\Psi (\vartheta )]\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(\eta ){\diamond }_{\frac{1}{2}}\eta +[\Phi (\zeta )\Psi (\vartheta )+\Phi (\vartheta )\Psi (\zeta )]\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta .\end{array}$$

By using $(1),$ $$\begin{array}{c}\left[\frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\underset{¯}{\Phi }({\varrho }_1)\underset{¯}{\Psi }({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1,\frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\overline{\Phi }({\varrho }_1)\overline{\Psi }({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1\right]\\ \supseteq \phi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(\eta ){\diamond }_{\frac{1}{2}}\eta +\chi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta .\\ \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varrho }_1)\Psi ({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1\supseteq \phi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(\eta ){\diamond }_{\frac{1}{2}}\eta +\chi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_1(\eta )h_2(1-\eta ){\diamond }_{\frac{1}{2}}\eta ,\end{array}$$ where $\phi (\zeta ,\vartheta )=\Phi (\zeta )\Psi (\zeta )+\Phi (\vartheta )\Psi (\vartheta )$ and $\chi (\zeta ,\vartheta )=\Phi (\zeta )\Psi (\vartheta )+\Phi (\vartheta )\Psi (\zeta ).$ □

Corollary 3.15.

If we choose time scale to be set of real numbers in Theorem 3.14, it becomes (Zhao et al., 2018, Theorem 4.5).

Corollary 3.16.

When $\mathbb{T}={\alpha }_1\mathbb{N}$ and $\zeta ={\alpha }_1,\vartheta =(n+1){\alpha }_1,{\alpha }_1>0$ in Theorem 3.14, set inclusion (22)(22) $$\frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varrho }_1)\Psi ({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1\supseteq \phi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta +\chi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(1-\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta ,$$(22) becomes $$\begin{array}{c}\sum _{l=2}^n\Phi (l{\alpha }_1)\Psi (l{\alpha }_1)+\frac{\Phi ({\alpha }_1)\Psi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)\Psi ((n+1){\alpha }_1)}{2}\hfill \\ \supseteq \phi ({\alpha }_1,(n+1){\alpha }_1)\sum _{l=2}^n{h}_1\left(\frac{l-1}{n}\right)h_2\left(\frac{l-1}{n}\right)+\frac{h_1(0)h_2(0)+h_1(1)h_2(1)}{2}\hfill \\ +\chi ({\alpha }_1,(n+1){\alpha }_1)\sum _{l=2}^n{h}_1\left(\frac{l-1}{n}\right)h_2\left(\right(\frac{n-l+1}{n}\left)\right)+\frac{h_1(0)h_2(1)+h_1(1)h_2(0)}{2}\hfill \end{array},$$ where $$\phi ({\alpha }_1,(n+1){\alpha }_1)=\Phi ({\alpha }_1)\Psi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)\Psi ((n+1){\alpha }_1),$$ and $$\chi ({\alpha }_1,(n+1){\alpha }_1)=\Phi ({\alpha }_1)\Psi ((n+1){\alpha }_1)+\Phi ((n+1){\alpha }_1)\Psi ({\alpha }_1).$$

Example 3.17.

Let ${\alpha }_1=3,n=2,\Phi ({\varrho }_1)=[\frac{1}{{\varrho }_1},\sqrt{{\varrho }_1}],\Psi ({\varrho }_1)=[{\varrho }_1,10\sqrt{{\varrho }_1}].$ In this case $\Phi ({\varrho }_1)\Psi ({\varrho }_1)=[1,10{\varrho }_1].$ Further, choose $h_1(\eta )=h_2(\eta )=\eta$ in Corollary 3.16 to get $$[1,60]\supset [1.16625,57.99].$$

Corollary 3.18.

When $\mathbb{T}=q^{\mathbb{N}}$ and ζ = 1, $\vartheta =q^{n+1},q>1$ in Theorem 3.14, then set inclusions (22)(22) $$\frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varrho }_1)\Psi ({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1\supseteq \phi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta +\chi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(1-\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta ,$$(22) take the form $$\begin{array}{c}\sum _{l=0}^n{q}^l\left(\Phi (q^l)\Psi (q^l)+\Phi (q^{l+1})\Psi (q^{l+1})\right)\\ \supseteq \phi (1,q^{n+1})\sum _{l=0}^n\left(\frac{q^l}{q^{n+1}-1}\right)\left(h_1\left(\frac{q^l-1}{q^{n+1}-1}\right)h_2\left(\frac{q^l-1}{q^{n+1}-1}\right)+h_1\left(\frac{q^{l+1}-1}{q^{n+1}-1}\right)h_2\left(\frac{q^{l+1}-1}{q^{n+1}-1}\right)\right)\\ +\chi (1,q^{n+1})\sum _{l=0}^n\left(\frac{q^l}{q^{n+1}-1}\right)\left(h_1\left(\frac{q^l-1}{q^{n+1}-1}\right)h_2\left(\frac{q^{n+1}-q^l}{q^{n+1}-1}\right)+h_1\left(\frac{q^{l+1}-1}{q^{n+1}-1}\right)h_2\left(\frac{q^{n+1}-q^{l+1}}{q^{n+1}-1}\right)\right)\end{array}$$ where $$\phi (1,q^{n+1})=\Phi (1)\Psi (1)+\Phi (q^{n+1})\Psi (q^{n+1}),$$ and $$\chi (1,q^{n+1})=\Phi (1)\Psi (q^{n+1})+\Phi (q^{n+1})\Psi (1).$$

Example 3.19.

Choose $\mathbb{T}=3^{{\mathbb{N}}_0},\left[\zeta ,\vartheta \right]=[1,10]$ In this case ${\left[\zeta ,\vartheta \right]}_{\mathbb{T}}=\left\{1,3,3^2\right\}.$ Also assume $\Phi ({\varrho }_1)=\left[\frac{1}{{\varrho }_1},\sqrt{{\varrho }_1},\right]$ $\Psi ({\varrho }_1)=\left[{\varrho }_1,10\sqrt{{\varrho }_1}\right],$ $h_1(\eta )=h_2(\eta )=1$ in Corollary 3.18 to get $$\left[1,50\right]\supseteq \left[1.388,20\right].$$

Theorem 3.20.

Let $\Phi ,\Psi :I_{\mathbb{T}}\to {\mathbb{R}}_I^{+}$ be two interval valued functions which are h-convex with $\Phi ({\varrho }_1)=\left[\underset{¯}{\Phi }({\varrho }_1),\overline{\Phi }({\varrho }_1)\right],\Psi ({\varrho }_1)=[\underset{¯}{\Psi }({\varrho }_1),\overline{\Psi }({\varrho }_1)]$ and $\Phi \Psi \in I{R}_{({[\zeta ,\vartheta ]}_{\mathbb{T}})},h_1,h_2:{\left[0,1\right]}_{\mathbb{T}}\to \mathbb{R}$ are positive right dense continuous functions and $h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\ne 0$. If $\Phi \in S{X}_1(h_1,I_{\mathbb{T}},{\mathbb{R}}_I^{+}),\Psi \in S{X}_1(h_2,I_{\mathbb{T}},{\mathbb{R}}_I^{+})$, then (23) $$\begin{array}{c}\frac{1}{2h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)}\Phi \left(\frac{\zeta +\vartheta }{2}\right)\Psi \left(\frac{\zeta +\vartheta }{2}\right)\\ \supseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varrho }_1)\Psi ({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1+\phi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta +\chi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(1-\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta .\end{array}$$(23)

Proof.

By hypothesis, one has that $$\begin{array}{c}{h}_1\left(\frac{1}{2}\right)\Phi \left((1-\eta )\vartheta +\eta \zeta \right)+h_1\left(\frac{1}{2}\right)\Phi \left(\eta \zeta +(1-\eta )\vartheta \right)\subseteq \Phi \left(\frac{\zeta +\vartheta }{2}\right)\\ h_2\left(\frac{1}{2}\right)\Psi \left((1-\eta )\vartheta +\eta \zeta \right)+h_2\left(\frac{1}{2}\right)\Psi \left(\eta \zeta +(1-\eta )\vartheta \right)\subseteq \Psi \left(\frac{\zeta +\vartheta }{2}\right).\end{array}$$

Then $$\begin{array}{c}\Phi \left(\frac{\zeta +\vartheta }{2}\right)\Psi \left(\frac{\zeta +\vartheta }{2}\right)\\ \supseteq h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)[\underset{¯}{\Phi }\left((1-\eta )\vartheta +\eta \zeta \right)\underset{¯}{\Psi }\left((1-\eta )\vartheta +\eta \zeta \right)+\underset{¯}{\Phi }\left((1-\eta )\vartheta +\eta \zeta \right)\underset{¯}{\Psi }\left(\eta \zeta +(1-\eta )\vartheta \right)\\ +\underset{¯}{\Phi }\left(\eta \zeta +(1-\eta )\vartheta \right)\underset{¯}{\Psi }\left((1-\eta )\vartheta +\eta \zeta \right)+\underset{¯}{\Phi }\left(\eta \zeta +(1-\eta )\vartheta \right)\underset{¯}{\Psi }\left(\eta \zeta +(1-\eta )\vartheta \right),\\ \overline{\Phi }\left((1-\eta )\vartheta +\eta \zeta \right)\overline{\Psi }\left((1-\eta )\vartheta +\eta \zeta \right)+\overline{\Phi }\left((1-\eta )\vartheta +\eta \zeta \right)\overline{\Psi }\left(\eta \zeta +(1-\eta )\vartheta \right)\\ +\overline{\Phi }\left(\eta \zeta +(1-\eta )\vartheta \right)\overline{\Psi }\left((1-\eta )\vartheta +\eta \zeta \right)+\overline{\Phi }\left(\eta \zeta +(1-\eta )\vartheta \right)\overline{\Psi }\left(\eta \zeta +(1-\eta )\vartheta \right)]\\ =h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[\underset{¯}{\Phi }\left((1-\eta )\vartheta +\eta \zeta \right)\underset{¯}{\Psi }\left((1-\eta )\vartheta +\eta \right),\overline{\Phi }\left((1-\eta )\vartheta +\eta \zeta \right)\overline{\Psi }\left((1-\eta )\vartheta +\eta \zeta \right)\right]\\ +h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[\underset{¯}{\Phi }\left((1-\eta )\vartheta +\eta \zeta \right)\underset{¯}{\Psi }\left(\eta \zeta +(1-\eta )\vartheta \right),\overline{\Phi }\left((1-\eta )\vartheta +\eta \zeta \right)\overline{\Psi }\left(\eta \zeta +(1-\eta )\vartheta \right)\right]\\ +h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[\underset{¯}{\Phi }\left(\eta \zeta +(1-\eta )\vartheta \right)\underset{¯}{\Psi }\left((1-\eta )\vartheta +\eta \zeta \right),\overline{\Phi }\left(\eta \zeta +(1-\eta )\vartheta \right)\overline{\Psi }\left((1-\eta )\vartheta +\eta \zeta \right)\right]\\ +h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[\underset{¯}{\Phi }\left(\eta \zeta +(1-\eta )\vartheta \right)\underset{¯}{\Psi }\left(\eta \zeta +(1-\eta )\vartheta \right),\overline{\Phi }\left(\eta \zeta +(1-\eta )\vartheta \right)\overline{\Psi }\left(\eta \zeta +(1-\eta )\vartheta \right)\right]\\ =h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[\Phi \left((1-\eta )\vartheta +\eta \zeta \right)\Psi \left((1-\eta )\vartheta +\eta \zeta \right)+\Phi \left(\eta \zeta +(1-\eta )\vartheta \right)\Psi \left(\eta \zeta +(1-\eta )\vartheta \right)\right]\end{array}$$ $$\begin{array}{c}+h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[\Phi \left((1-\eta )\vartheta +\eta \zeta \right)\Psi \left(\eta \zeta +(1-\eta )\vartheta \right)+\Phi \left(\eta \zeta +(1-\eta )\vartheta \right)\Psi \left((1-\eta )\vartheta +\eta \zeta \right)\right]\\ \supseteq h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[\Phi \left((1-\eta )\vartheta +\eta \zeta \right)\Psi \left((1-\eta )\vartheta +\eta \zeta \right)+\Phi \left(\eta \zeta +(1-\eta )\vartheta \right)\Psi \left(\eta \zeta +(1-\eta )\vartheta \right)\right]\\ +h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[(h_1((1-\eta ))\Phi (\vartheta )+h_1\eta \Phi (\zeta )\right)\left(h_2(\eta )\Psi (\zeta )+h_2(1-\eta )\Psi (\vartheta )\right)\\ +\left(h_1(\eta )\Phi (\zeta )+h_1(1-\eta )\Phi (\vartheta )\right)\left(h_2(1-\eta )\Psi (\vartheta )+h_2(\eta )\Psi (\zeta )\right)]\\ =h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[\Phi \left((1-\eta )\vartheta +\eta \zeta \right)\Psi \left((1-\eta )\vartheta +\eta \zeta \right)+\Phi \left(\eta \zeta +(1-\eta )\vartheta \right)\Psi \left(\eta \zeta +(1-\eta )\vartheta \right)\right]\\ +h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)\left[(h_1(1-\eta )h_2(\eta )+h_1(\eta )h_2(1-\eta )\right)\phi (\zeta ,\vartheta )\\ +\left(h_1(1-\eta )h_2(1-\eta )+h_1(\eta )h_2(\eta )\right)\chi (\zeta ,\vartheta )].\end{array}$$

Integration over [0,1], gives $$\begin{array}{c}\frac{1}{2h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)}\Phi \left(\frac{\zeta +\eta }{2}\right)\Psi \left(\frac{\zeta +\eta }{2}\right)\\ \supseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varrho }_1)\Psi ({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1+\phi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta +\chi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(1-\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta .\end{array}$$

It concludes the proof. □

Corollary 3.21.

If we choose time scale to be set of real numbers in Theorem $3.20$, it becomes (Zhao et al., 2018, Theorem 4.6).

Corollary 3.22.

When $\mathbb{T}={\alpha }_1\mathbb{N}$ and $\zeta ={\alpha }_1,\eta =(n+1){\alpha }_1,{\alpha }_1>0$ in Theorem 3.20, then set inclusion (23)(23) $$\begin{array}{c}\frac{1}{2h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)}\Phi \left(\frac{\zeta +\vartheta }{2}\right)\Psi \left(\frac{\zeta +\vartheta }{2}\right)\\ \supseteq \frac{1}{\vartheta -\zeta }\underset{\zeta }{\overset{\vartheta }{\int }}\Phi ({\varrho }_1)\Psi ({\varrho }_1){\diamond }_{\frac{1}{2}}{\varrho }_1+\phi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta +\chi (\zeta ,\vartheta )\underset{0}{\overset{1}{\int }}{h}_2(1-\eta )h_1(\eta ){\diamond }_{\frac{1}{2}}\eta .\end{array}$$(23) becomes $$\begin{array}{c}\frac{n}{2h_1\left(\frac{1}{2}\right)h_2\left(\frac{1}{2}\right)}\Phi \left(\frac{{\alpha }_1+(n+1){\alpha }_1}{2}\right)\Psi \left(\frac{{\alpha }_1+(n+1){\alpha }_1}{2}\right)\\ \supseteq \sum _{l=2}^n\Phi (l{\alpha }_1)\Psi (l{\alpha }_1)+\frac{\Phi ({\alpha }_1)\Psi ({\alpha }_1)+\Phi ((n+1){\alpha }_1)\Psi ((n+1){\alpha }_1)}{2}\\ +\phi ({\alpha }_1,(n+1){\alpha }_1)\sum _{l=2}^n{h}_1\left(\frac{l-1}{n}\right)h_2\left(\frac{l-1}{n}\right)+\frac{h_1(0)h_2(0)+h_1(1)h_2(1)}{2}\\ +\chi ({\alpha }_1,(n+1){\alpha }_1)\sum _{l=2}^n{h}_1\left(\frac{l-1}{n}\right)h_2\left(\right(\frac{n-l+1}{n}\left)\right)+\frac{h_1(0)h_2(1)+h_1(1)h_2(0)}{2}.\end{array}$$

Example 3.23.

Let ${\alpha }_1=3,n=2,\Phi ({\varrho }_1)=[\frac{1}{{\varrho }_1},\sqrt{{\varrho }_1}],\Psi ({\varrho }_1)=[{\varrho }_1,10\sqrt{{\varrho }_1}].$ In this case $\Phi ({\varrho }_1)\Psi ({\varrho }_1)=[1,10{\varrho }_1].$ Further, choose $h_1(\eta )=h_2(\eta )=\eta$ in Corollary 3.22 to get $$[2,120]\supset [2.16625,117.99].$$

4. Conclusion

We have studied the h-convex (affine, concave) functions for interval valued functions. We proved Jensen’s type inequalityin for interval valued h-convex functions on time scales. In seek of applications, we proved interval valued Hermite-Hadamard type inequalities on time scales. Present results generalize the existing inequalities given in Zhao et al. (2018). We have discussed all the obtained inequalities by choosing $\mathbb{T}=\alpha \mathbb{Z}$ for $(\alpha >0).$ Which are also new up to knowledge of authors. Some numerical examples are also provided to illustrate the results.

Disclosure statement

No potential conflict of interest was reported by the authors.