Refinements on the discrete Hermite–Hadamard inequality

In this paper, we use techniques and tools from time scale calculus to state and prove many refinements on the discrete Hermite–Hadamard inequality.


Introduction
The Hermite-Hadamard inequality [9,10] states that if f : I → R is a convex function, then the following inequality is satisfied: where a, b ∈ I and I is an interval in R. Hermite-Hadamard's inequality for convex functions has received renewed attention in recent years and a remarkable variety of refinements and generalizations have been found (see, for example, [3][4][5][6]12,13]). Regarding the definition of a convex function on a time scale, a pioneering work has been done by Mozyrska and Torres [11]. They introduced the convexity of a function defined on a time scale (a nonempty closed subset of R) and proved a theorem which characterizes the convex function in terms of its second derivative.
Since the midpoint condition plays an important role in the proofs related to the Hermite-Hadamard inequality, the authors first defined the midpoint condition for the functions on the set of integers, Z, in [1]. Then, some equivalent conditions for convexity have been given, and the discrete Hermite-Hadamard inequality has been proven.
We note that T [a,b] is a subset of the real interval [0, 1]. Since the set T [a,b] is an isolated time scale, from now on we use the notations of the time scale calculus. Definition 1.1 [1] f : Z → R is called convex on Z if for every x, y ∈ Z with x < y the following inequality is satisfied for all λ ∈ T [x,y] .
Z with a, b ∈ Z and a < b, and a + b an even number. Then Our aim in this paper is to continue our work on the discrete Hermite-Hadamard inequality. We state and prove some refinements on both sides of the inequality (4).
The plan of the paper is as follows: In Sect. 2, we list the integration by parts formulas and the substitution rules for integrals on time scales. For the purpose of comparison, the continuous case and the discrete case, we list the refinements obtained for the continuous Hermite-Hadamard inequality. In Sect. 3, we state and prove some refinements related to both sides of the discrete Hermite-Hadamard inequality.

Preliminaries
The following two theorems will play important roles in the proof of our main results.
(ii) Assume ν is strictly decreasing and T : −ν (T) is a time scale. If f : T → R is an rd-continuous function and ν is differentiable with rd-continuous derivative, In [7], Dragomir and Agarwal proved the following results connected with the right part of (1). L [a, b], then the following equality holds: , then the following equality holds:

Main results
In this section, we start with stating and proving the discrete counterparts of Lemma 2.3 and Theorem 2.1. Then, some other refinements will follow.
To prove our first claim, we first use substitution method given in (Theorem 2.2) and then use integration by parts formula (Theorem 2.1 ). Here, we have ν (t) = b−t b−a . We apply the substitution method (Theorem 2.2) for the integral 1 (t)). Now, we apply integration by parts to the last integral, we have If we multiply the last quantity by b−a 4 , we have the desired result. Next, we claim that To prove this claim, we first use the substitution rule (Theorem 2.2) and then use integration by parts formula (Theorem 2.1). Here, we have where ν(t) = t−a b−a . Next, we use the integration by parts formula to the last integral above. We have Adding (7) and (8) Proof Using Lemma (3.1) and the convexity on [a, b] Z of f and f ∇ , we obtain  Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.