New Estimates for Hermite-Hadamard Inequality in Quantum Calculus via (α, m) Convexity

Abstract

This study provokes the existence of quantum Hermite-Hadamard inequalities under the concept of q-integral. We analyse and illustrate a new identity for the differentiable function mappings whose second derivatives in absolute value are (α,m) convex. Some basic inequalities such as Hölder’s and Power mean have been used to obtain new bounds and it has been determined that the main findings are generalizations of many results that exist in the literature. We make links between our findings and a number of well-known discoveries in the literature. The conclusion in this study unify and generalise previous findings on Hermite-Hadamard inequalities.

On q-Hermite–Hadamard Type Inequalities via s-Convexity and (α,m)-Convexity

“…In [20], the authors presented new estimates for q-Hermite-Hadamard inequality for mappings whose second q-derivatives in absolute value are (α, m)-convex in Theorems 4-7, which are demonstrated below. Similar inequalities will be provided in the next section for functions whose third q-derivatives in absolute value are s-convex.…”

“…We will provide new q-Hermite-Hadamard-like type inequalities analogue to the same from Theorems 4-7 from [20], which it is then necessary to recall below, for the case of mappings whose third derivatives in absolute values are s-convex. In order to obtain new q-Hermite-Hadamard-like type inequalities for mappings whose third derivatives in absolute values are s-convex, a new quantum identity will be established below.…”

“…The concept of convexity is used for solving problems in mathematics, which have many applications in other domains such as business and industry. Hermite-Hadamard inequality (H-H) [7] is one of the most well-known inequalities from the theory of convexity and have attracted a lot of attention; see the large number of refinements [8][9][10] and generalizations [11][12][13], as for example [14][15][16][17][18][19] without using q-calculus, and in the frame of q-calculus and (p, q)-calculus, see for example, [20][21][22][23][24][25][26]. In [14,17], for example, different types of Hermite-Hadanmard inequalities were presented for functions whose second derivatives in absolute value satisfy different types of convexities using in the equality of the corresponding key lemma two integrals from 0 to 1.…”

“…Inspired by the recent work of [20,21], respectively, our first purpose is to find new q-Hermite-Hadamard like type inequalities for three time q-differentiable s-convex functions by using a new auxiliary tool, a q-integral identity for the q-left and the q-right derivatives of order three. Our second purpose is to present new q-Hermite-Hadamard like type inequalities with two parameters for twice q-differentiable (α, m)-convex functions, for the q-left and q-right derivatives of order two.…”

New Perspectives of Symmetry Conferred by q-Hermite-Hadamard Type Integral Inequalities

“…Moreover, it seems to have "a pivotal role in several pure and applied sciences" [1]. Integral inequalities, on the other hand, "plays a vital role in the theory of differential equations" [2]. The very famous Hermite-Hadamard inequality is a part of integral inequalities and have been intensively studied by numerous scholars in the last decades.…”