Symmetrically _γ-Convex Functions

Abstract

Given γ>0, a real-valued function f defined on a noempty convex set D of a normed space X is said to be symmetrically γ-convex iff for all x _oand X ₁∊D satisfying , the Jensen inequality

is fulfilled for and for . Though the Jensen inequality is only required to hold true at two points and on the segment [x ₀,x ₁] satisfying , such a function has interesting properties similar to those of classical convex functions. For instance, a symmetrically γ-convex function from a finite-dimensional normed space X is locally Lipschitzian at each γ-interior point of D, i.e., at each point x∊D for which there is some ε>0 such that implies . This and other properties of this function class are established in our paper

Keywords:

• Generalized Convexity
• Rough Convexity
• γ-Convexity
• Locally Lipschitz Continuity
• Differentiability

^∗ Corresponding author

^∗ Corresponding author

Notes

^∗ Corresponding author