Conditional Convexity

Abstract

It is well known that a function f:D→R Fréchet differentiable on an open convex subset D of a real normed linear space is convex, i.e.,fλx+1−λy≤λfx+1−λfyx,y∈D,λ∈0,11holds if and only iff′xy−x≤fy−fxx,y∈I2is valid [see, e.g., Roberts and Varberg (“Convex Functions,” Academic Press, New York and London, 1973)].It is shown that (1) with a fixedy=w (or with fixed λx+(1−λ)y=w) is equivalent to the inequality (2) with fixedy=w (or with fixedx=w, respectively).Then these results are applied to study some conditional inequalities for deviation means.