Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus

Sitho S., Ali M. A., Budak H., Ntouyas S. K., Tariboon J.

MATHEMATICS, vol.9, no.14, 2021 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 9 Issue: 14
  • Publication Date: 2021
  • Doi Number: 10.3390/math9141666
  • Journal Name: MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Communication Abstracts, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Kocaeli University Affiliated: No

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(x)(2)- quantum integral to show midpoint and trapezoidal inequalities for q(x)(2)-differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite- Hadamard-type inequality for preinvex functions via q(x)(1)-quantum integral is not valid for preinvex functions, and we present its proper form. We use q(x)(1)-quantum integrals to show midpoint inequalities for q(x)(1)-differentiable preinvex functions. It is also demonstrated that by considering the limit q -> 1(-) and eta(x(2), x(1)) = -eta(x(1), x(2)) = x(2), x(1) in the newly derived results, the newly proved findings can be turned into certain known results.