Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus

Surang Sitho, Muhammad Aamir Ali, Hüseyin Budak, Sotiris K. Ntouyas and Jessada Tariboon
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Surang Sitho: Department of Social and Applied Science, College of Industrial Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
Muhammad Aamir Ali: Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
Hüseyin Budak: Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce 81620, Turkey
Sotiris K. Ntouyas: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Jessada Tariboon: Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand

Mathematics, 2021, vol. 9, issue 14, 1-21

Abstract: In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the q ? 2 -quantum integral to show midpoint and trapezoidal inequalities for q ? 2 -differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via q ? 1 -quantum integral is not valid for preinvex functions, and we present its proper form. We use q ? 1 -quantum integrals to show midpoint inequalities for q ? 1 -differentiable preinvex functions. It is also demonstrated that by considering the limit q ? 1 ? and ? ? 2 , ? 1 = ? ? ? 1 , ? 2 = ? 2 ? ? 1 in the newly derived results, the newly proved findings can be turned into certain known results.

Keywords: Hermite–Hadamard inequality; q-integral; quantum calculus; preinvex function; trapezoid inequalities; midpoint inequalities (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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