Ostrowski type inequalities for exponentially s-convex functions on time scale

1.   Introduction

Ostrowski formulate a formula in 1938 to calculate the deviation of differentiable functions from its integral mean which is discussed in ^[1] and known as Ostrowski inequality given by

can be proved by Montgomery identity as shown in ^[2] but this identity on time scale was discussed by M. Bohner and T. Matthews in ^[3] which is given as

Lemma 1.1. Let $l_1, l_2, u, v \in T, l_1 < l_2$ and $\xi:[l_1, l_2]\to R$ be differentiable. Then

Where

Definition 1.1. (^[4]) Let $s\in (0, 1]$. A function $\xi:I\subseteq R_{0}\to R_{0}$, where $R_{0} = [0, \infty)$, is said to be s-convex function in second sense if

for all $l_{1}, l_{2}\in I$.

Definition 1.2. (^[5]) Let $s\in (0, 1]$. A function $\xi:I\subseteq R \to R$ is said to be exponentially convex if

for $l_{1}, l_{2} \in I$ with $l_{1} < l_{2}$, $k\in[0, 1]$ and $\alpha\in R$.

Our aim of this paper is to discuss Hermite Hadamard inequality and Ostrowski type inequalities on time scale for exponentially $s$-convex, $s$-convex functions.

2.   Preliminaries

Definition 2.1. Time scale is defined as a non-empty close subset of real numbers.

The most important examples are $R$ (set of real numbers) and $Z$ (set of integers). For $u, v \in T$ where $T$ is a time scale, forward and backward jumped operators $\sigma$ and $\rho$ respectively are defined as $\sigma(v) = inf\{k\in T:k > v\}\in T$, $\rho(v) = sup\{k\in T:k < v\}\in T$. Supplemented by $inf \phi = sup T$ and $sup \phi = inf T$.

A point $v$ is said to be right scattered and left scattered if $\sigma(v) > v$ and $\rho(v) < v$ respectively. If a point $v$ is both right and left scattered then it is isolated. If $\sigma(v) = v$ then $v$ is called right dense and it is said to be left dense if $\rho(v) = v$. If the point $v$ is left and right dense both then it is called dense.

Suppose $\zeta_1\in T$ is right scattered minimum, then $T_k = T-\{\zeta_1\}$ otherwise $T_k = T$. Suppose $\zeta_2\in T$ is left scattered maximum, then $T^k = T-\{\zeta_2\}$, otherwise $T^k = T$. Moreover $T_{k}^k = T_k \cap T^k$.

Definition 2.2. Delta derivative of function $\xi:T\to R$ at $v\in T^k$ is defined to be the number $\xi^\Delta(v)$ (if it exists) satisfying the property that, for any $\epsilon > 0$ there is a neighbourhood $U$ of $v$ such that

for all $u\in U$.

Definition 2.3. A function $\xi:T\to R$ is continuous at right dense points of $T$ and its left-sided limit exist at left dense points of $T$, then $\xi$ is known to be $rd$-continuous. Denoted by $\xi\in C_{rd}$.

Theorem 2.1. Let $\xi:T\to R$ be an rd-continuous function. Then $f$ has an anti-derivative $\Xi$ satisfying $\Xi^\Delta = \xi$.

Proof. See [6,Theorem 1.74].

Definition 2.4. If $\xi:T \to R$ is an rd-continuous function and $l_1 \in T$, then we define the integral $\Xi(v) = \int_{l_1}^{v} \xi(k)\Delta k$ for $v \in T$.

Therefore for $\xi\in C_{rd}$ we have $\Xi(l_2)-\Xi(l_1) = \int_{l_1}^{l_2} \xi(k)\Delta k$. Where $\Xi^\Delta = \xi$.

Theorem 2.2. If $l_1, l_2, l_3 \in T, \beta \in R$ and $\xi_1, \xi_2\in C_{rd}$, then

(i) $\int_{l_1}^{l_2}(\xi_1(v)+\xi_2(v))\Delta v = \int_{l_1}^{l_2} \xi_1(v)\Delta v+\int_{l_1}^{l_2} \xi_2(v)\Delta v$,

(ii) $\int_{l_1}^{l_2} \beta \xi_1(v)\Delta v = \beta\int_{l_1}^{l_2} \xi_1(v)\Delta v$,

(iii) $\int_{l_1}^{l_2} \xi_1(v)\Delta v = -\int_{l_2}^{l_1} \xi_1(v)\Delta v$,

(iv) $\int_{l_1}^{l_2} \xi_1(v)\Delta v = \int_{l_1}^{l_3} \xi_1(v)\Delta v +\int_{l_3}^{l_2} \xi_1(v)\Delta v$,

(v) $\int_{l_1}^{l_1} \xi_1(v)\Delta v = 0$,

(vi)$\int_{l_1}^{l_2}\xi_1(v)\xi_2^{\Delta}(v)\Delta v = (\xi_1\xi_2)(l_2)-(\xi_1\xi_2)(l_1)-\int_{l_1}^{l_2}\xi_1^{\Delta}(v)\xi_2(\sigma(v))\Delta v$,

(vii)$\int_{l_1}^{l_2} \xi_1(v) \xi_2^\Delta(v) = (\xi_1 \xi_2)(l_2)-(\xi_1 \xi_2)(l_1)-\int_{l_1}^{l_2} \xi_1^\Delta(v) \xi_2(\sigma (v))\Delta v$.

Proof. See [6,Theorem 1.77].

Theorem 2.3. Let $l_1, l_2\in T$ and $\xi_1, \xi_2:T\to R$ be rd-continuous. Then

where $p$, $q > 1$ and $\frac{1}{p}+\frac{1}{q} = 1$.

Proof. See [66.Theorem 6.13].

3.   Main results

Keeping in mind the integral inequalities and inequalities on time scale ^{[7,8,9,10,11,12,13,14,15]} first we prove the Hermite Hadamard inequality for exponentially s-convex functions on time scale. Throughout this section $K = [l_{1}, l_{2}]\subseteq T$.

Theorem 3.1. Let $T$ be a time scale and $K = [l_1, l_2]$. Let $\xi:K\to R$ is exponentially s-convex function in the second sense on $K^0$ and $\Delta$-integrable as well. Then for $l_1, l_2 \in K$ with $l_1 < l_2$ and $\alpha \in R$, we have

Proof. Using the definition of exponential s-convexity of $\xi$ we have

Making use of change of variable $x = kl_1+(1-k)l_2$ and $y = (1-k)l_1+kl_2$ and taking $\Delta$-integral with respect to $k\in[0, 1]$ we get

and

Now, we prove second inequality

Taking $\Delta$-integral w.r.t $k\in [0, 1]$ we get

Combining (7) and (8) we get (6).

Corollary 3.1.1. For $T = R$ we get the Hermite Hadamard inequality for exponentially s-convex functions [5,Theorem 3.2].

Now, we will discuss Ostrowski inequality for exponentially s-convex function on time scale.

Theorem 3.2. Let $T$ be a time scale and $K\subseteq T$. Let $\xi:K\to R$ be a differentiable function on $K^0$ such that $\xi^\Delta\in K$ for $l_1, l_2\in K$ where $l_1 < l_2$. If $\xi^\Delta$ is exponentially s-convex in second sense on $[l_1, l_2]$ for $s\in(0, 1]$ and $\sup_{l_1\le v\le l_2}|\xi^\Delta(v)| = M$, $v\in [l_1, l_2]$. Then following inequality holds:

Proof. Using Montgomery identity

Making use of change of variables we get

Using exponential s-convexity of $\xi^\Delta$ we get

Corollary 3.2.1. If $T = R$, then we obtain Theorem 2.1 given in ^[16].

Theorem 3.3. Suppose that $\xi:K \to R$ be a differentiable mapping on $K^0$ such that $\xi^\Delta \in K$ for $l_1, l_2\in K$ with $l_1 < l_2$. If $|\xi^\Delta|^q$ is exponentially s-convex in the second sense on $[l_1, l_2]$ for some $s\in (0, 1],$ $p, q > 1$ and $\frac{1}{p}+\frac{1}{q} = 1$ and $\sup_{l_1\le v\le l_2}|\xi^\Delta (v)| = M, v\in[l_1, l_2]$, then the following inequality holds:

Proof. By Montgomery identity we have

Making use of change of variables we obtain

Using (5) we get

Using the definition of exponential s-convexity of $|\xi^\Delta|^q$ we have

Corollary 3.3.1. If $T = R$ then we obtain Theorem 2.2 given in ^[16].

Theorem 3.4. Let us consider a differentiable mapping $\xi:K \to R$ on $K^0$ such that $\xi^\Delta \in K$ for $l_1, l_2\in K$ with $l_1 < l_2$. If $|\xi^\Delta|^q$ is exponentially s-convex in the second sense on $[l_1, l_2]$ for some $s\in (0, 1]$, $q > 1$ and $\sup_{l_1\le v\le l_2}|\xi^\Delta (v)| = M,$ $v\in[l_1, l_2]$, then the following inequality holds:

Proof. By Montgomery identity we have

Making use of change of variables we obtain

It follows that

Using the definition of exponential s-convexity of $|\xi^\Delta|^q$ we have

Corollary 3.4.1. If $T = R$ then we obtain Theorem 2.3 given in ^[16].

Theorem 3.5. Let $\xi:K\to R$ be a differentiable mapping on $K^0$ such that $\xi^\Delta\in K$ for $l_1$, $l_2\in K$ with $l_1 < l_2$. If $|\xi^\Delta|^q$ is exponentially s-concave on $[l_1, l_2]$, $p > 1$, $\frac{1}{p}+\frac{1}{q} = 1$, then the following inequality holds:

Proof. Using Montgomery identity and making use of variables we get

Since $|\xi^\Delta|^q$ is exponentially s-concave, by (6) we have

and

Using (14) and (15) in (13) we get the conclusion.

Corollary 3.5.1. If $T = R$ then we obtain Theorem 2.4 given in ^[16].

Now we discuss some results for s-convex functions.

Theorem 3.6. Let $T$ be a time scale and $K = [l_1, l_2]\subseteq T$ such that $l_1 < l_2\in T$. Let $\xi:K\to R$ be a delta differentiable on $K^0$ such that $\xi^\Delta \in K$, for $l_1, l_2\in K$ with $l_1 < l_2$. If $|\xi^\Delta|$ is s-convex on $K$ for some fixed $s\in (0, 1]$ and $\sup_{l_1\le v\le l_2}\left|\xi^\Delta(v)\right| = M$ for $v\in K$, then following inequality holds:

Proof. The proof is analogous to Theorem 3.2 only difference is to use definition of s-convex function $|\xi^\Delta|$ instead of exponentially s-convexity.

Corollary 3.6.1. If $T = R$, then we obtain Theorem 2 given in ^[17].

Theorem 3.7. Let $T$ be a time scale and $K = [l_1, l_2]\subseteq T$ such that $l_1 < l_2\in T$. Let $\xi:K\to R$ be a delta differentiable on $K^0$ such that $\xi^\Delta \in K$, for $l_1, l_2\in K$ with $l_1 < l_2$. If $|\xi^\Delta|^q$ is s-convex on $K$ for some fixed $s\in (0, 1]$, $p$, $q > 1$, $\frac{1}{p}+\frac{1}{q} = 1$ and $\sup_{l_1\le v\le l_2}\left|\xi^\Delta(v)\right| = M$ for $v\in K$, then following inequality holds:

Proof. Proof is analogous to Theorem 3.3 but in place of definition of exponential s-convexity we use s-convexity of $|\xi^\Delta|^q$.

Corollary 3.7.1. If $T = R$, then we obtain Theorem 3 given in ^[17].

Theorem 3.8. Let $T$ be a time scale and $K = [l_1, l_2]\subseteq T$ such that $l_1 < l_2\in T$. Let $\xi:K\to R$ be a delta differentiable on $K^0$ such that $\xi^\Delta \in K$ for $l_1$, $l_2\in K$ with $l_1 < l_2$. If $|\xi^\Delta|^q$ is s-convex in second sense on $K$ for some fixed $s\in (0, 1]$, $q > 1$ and $\sup_{l_1\le v\le l_2}\left|\xi^\Delta(v)\right| = M$ for $v\in K$, then following inequality holds:

Proof. Proof is analogous to Theorem 3.4 but we use definition of s-convexity of $|\xi^\Delta|^q$ instead of exponential s-convexity.

Corollary 3.8.1. If $T = R$, then we obtain Theorem 4 given in ^[17].

4.   Conclusions

From Theorem 3.1 we obtain the Hermite-Hadamard inequality for exponentially s-convex functions on time scale. From Theorems 3.2–3.5 we obtain Ostrowski type inequalities for exponentially s-convex functions on time scale. From Theorems 3.6–3.8 we obtain Ostrowski type inequalities for s-convex functions on time scale.

Acknowledgments

This research article is supported by National University of Sciences and Technology(NUST), Islamabad, Pakistan.

Conflict of interest

The authors declare that there is no interest regarding the publication of this paper.