  Citation: Xu, P.; Ihsan Butt, S.; Ain, Q.U.; Budak, H. New Estimates for Hermite-Hadamard Inequality in Quantum Calculus via (α, m) Convexity. Symmetry 2022, 14, 1394. https://doi.org/10.3390/ sym14071394 Academic Editor: Palle E.T. Jorgensen Received: 19 June 2022 Accepted: 5 July 2022 Published: 6 July 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article New Estimates for Hermite-Hadamard Inequality in Quantum Calculus via ( α, m) Convexity Peng Xu 1 , Saad Ihsan Butt 2, * , Qurat Ul Ain 2 and Hüseyin Budak 3 1 School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Duyun 558000, China; gdxupeng@gzhu.edu.cn 2 Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan; quratulain4566@gmail.com 3 Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce 81620, Turkey; hsyn.budak@gmail.com * Correspondence: saadihsanbutt@gmail.com 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. Keywords: quantum calculus; Hermite-Hadamard inequalities; (α, m) convexity MSC: 26A33; 26D15; 26E60 1. Introduction When there is no limit in calculus, it is referred as q-calculus. Euler is the inventor of q-parameter and also the creator of q-calculus. Jackson began his work in a symmetrical manner in the nineteenth century and presented q-definite integrals. Q-calculus is used in a wide range of subjects, including mathematics, number theory, hyper geometry and physics. One can see in [1–4] and references therein. In q-calculus, we substitute classical derivative with difference operator, allowing you to work with sets of non-differentiable functions. Quantum difference operators are of tremendous importance because of their applications in a variety of mathematical disciplines, including orthogonal polynomials, basic hypergeometric functions, combinatorics, mechanics and the theory of relativity. Many essential concepts of quantum calculus are covered in Kac and Cheung’s book [5]. These ideas help us to develop new inequalities, which can be useful in the discovery of new boundaries. Integral inequalities is historically viewed as a classical field of research. From classical to modern applications, inequalities have been used in mathematical analysis. In 1934, Polya and Hardy introduced classical work on inequalities. Integral inequalities plays vital role in differential equation theory. Many researchers have studied integral inequalities in classical calculus along with their applications (see [6–9]) . Because the value of mathe- matical inequalities was well established in past, inequalities such as Hermite-Hadamard, Popoviciu’s, Steffensen-Grüss, Jensen, Hardy and Cauchy-Schwarz performed an essential role in the theory of classical calculus and q-calculus [10–14]. In convexity theory, Hermite-Hadamard is one of the most well known inequality, which was developed by Hermite and Hadamard (see also [15], [16] p. 137). Convexity is very simple and natural concept to solve many problems of mathematics. Convexity is Symmetry 2022, 14, 1394. https://doi.org/10.3390/sym14071394 https://www.mdpi.com/journal/symmetry