A New Characterization of E-Convex Functions

Recently a new criterion of quasi-semi-E-convex functions was introduced by Peng in 2006 for a new criterion of quisi-semi-E-convex functions.In this paper,firstly,we propose a necessary and sufficient condition for E-convex functions by using the definitions of E-convex function and convex function,that is,Let E:Rn→Rn,M■Rn is a E-convex set,E(M) is a convex set,f is a real-valued function defined in M,so f is E-convex function if and only if ф(λ)=f[E(y)+λ(E(x)-E(y))] is convex function under some conditions.Then we have the other theorem,let f ∶M→R is a upper semi-continuous function in convex set M■Rn,if there exists a linear mapping E:Rn→Rn,such that E(M)■M,for x∈M have f(x)≤f(E(x)),and yn→y as E(yn)→E(y),then f is a quasi-semi-E-convex function彐β∈(0,1)(or β=0,1),such that f(βE(x)+(1-β)E(y))≤max{f(x),f(y)},x,y∈M.