Characterizations of uniform convexity for differentiable functions

Abstract

We consider convex functions in d real variables. For applications, for example in optimization, various strengthened forms of convexity have been introduced. Among them, uniform convexity is one of the most general, defined by involving a so-called modulus φ. Inspired by three classical characterizations of ordinary convexity, we aim at characterizations of uniform convexity by conditions in terms of the gradient or the Hessian matrix of the considered function for certain classes of moduli φ.

Properties of higher order preinvex functions

“…The optimality conditions of the differentiable higher order strongly preinvex functions can be characterized by a class of higher order variational-like inequalities, which appears to be a new ones. Our results represent a significant refinement and improvement of the results of Alabdali et al [1] and Lin and Fukushima [10] and include the results of Mohsen et al [12] as special cases. As novel and innovative applications of these higher order strongly preinvex functions, we have obtained the parallelogram-like laws for uniformly Banach spaces.…”

“…Strongly convex functions were introduced and studied by Polyak [20], which played an important part in the optimization theory and related areas. For the applications of strongly convex functions in various fields of pure and applied sciences, for example, see [1,8,9,10,12,13,15,17,20,21,25] and the references therein. Lin and Fukushima [10] introduced the concept of higher order strongly convex functions and used it in the study of mathematical program with equilibrium constraints.…”

“…Higher order strongly generalized convex functions play an important role in many fields such as engineering design, economic equilibrium and multilevel game. Noor and Noor [17,19], Mohsen et al [12] and Alabdali et al [1] introduced some new classes of higher order strongly convex functions and studied their characterizations.…”

Generalized Higher Order Preinvex Functions and Equilibrium-like Problems

“…Mohsin et al [32] study higher-order convex functions which represent a significant improvement to the concepts for higher-order convex functions that are studied by Alabdali et al [33], Lin et al [34], Mako et al [35], and Olbrys [36].…”

Higher-Order Strongly Preinvex Fuzzy Mappings and Fuzzy Mixed Variational-Like Inequalities

“…HOS-preinvex functions and their generalization have many applications in different areas such as multilevel games, engineering design, and economical equilibrium. Some of the authors presented different types of HOS-preinvex functions and discussed their characterizations like Alabdali et al [18], Noor and Noor [9,19,20], and Mohsen et al [21] considered different classes and prove that the optimality condition of HOS-preinvex functions can be distinguished by different kinds of variational inequalities like strongly variational inequality, strongly variational-like inequality, HOS-variational inequality, and HOS-variational-like inequality.…”