q -Hermite–Hadamard Inequalities for Generalized Exponentially s , m ; η -Preinvex Functions

Abstract

In this article, we introduce a new extension of classical convexity which is called generalized exponentially $\left(s,m;\eta \right)$ -preinvex functions. Also, it is seen that the new definition of generalized exponentially $\left(s,m;\eta \right)$ -preinvex functions describes different new classes as special cases. To prove our main results, we derive a new ${}^{m{\kappa }_2}q$ -integral identity for the twice ${}^{m{\kappa }_2}q$ -differentiable function. By using this identity, we show essential new results for Hermite–Hadamard-type inequalities for the ${}^{m{\kappa }_2}q$ -integral by utilizing differentiable exponentially $\left(s,m;\eta \right)$ -preinvex functions. The results presented in this article are unification and generalization of the comparable results in the literature.