Copyright © 2012 Xue Wen Liu et al. 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

A class of -semipreinvex functions, which are some generalizations of the semipreinvex functions, and the -convex functions, is introduced. Examples are given to show their relations among -semipreinvex functions, semipreinvex functions and -convex functions. Some characterizations of -semipreinvex functions are also obtained, and some optimality results are given for a class of -semipreinvex functions. Ours results improve and generalize some known results.