Copyright © 2012 Xiaoling Liu and D. H. Yuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We firstly construct a concrete semi-invex set which is not invex. Basing on concept of semi-invex set, we introduce some kinds of generalized convex functions, which include semi--preinvex functions, strictly semi--preinvex functions and explicitly semi--preinvex functions. Moreover, we establish relationships between our new generalized convexity and generalized convexity introduced in the literature. With these relationships and the well-known results pertaining to common generalized convexity, we obtain results for our new generalized convexities. We extend the existing results in the literature.