On ω-quasiconvex functions

In the paper we introduce convexity-like notions based on modification of quasiconvexity. DEFINITION. Let I be a real interval and ω 0 a given number. We say that a function f : I → R is ω -quasiconvex, ω -quasiconcave, respectively, if f (tx+(1− t)y) max( f (x), f (y))−ωmin(t,1− t)|x− y|, f (tx+(1− t)y) max( f (x), f (y))−ωmax(t,1− t)|x− y|, for x,y ∈ I,t ∈ (0,1). If f : I → R is simultaneously ω -quasiconvex and ω -quasiconcave then we say that f is ω -quasiaffine. We characterize these notions, in particular we show that ω -quasiconcave functions coincide with Lipschitz functions with constant ω . We conclude the paper with the following separation type result. THEOREM. Let f : I → R be ω -quasiconvex function and g : I →R ω -quasiconcave such that f g . Then there exists an ω -quasiaffine function h : I → R such that f h g . Mathematics subject classification (2010): 26B25, 39B62.