Conditional Convexity1

It is well known that a function f : D Fr´echet differentiable on an open convex subset D of a real normed linear space is convex, i.e., fŽ x Ž1 .y. fŽx. Ž1 .fŽy. Žx, y D, Ž0, 1.. Ž1. holds if and only if f Žx.Žy x. fŽy. fŽx. Žx, y I. Ž2. is valid see, e.g., Roberts and Varberg Ž‘‘Convex Functions,’’ Academic Press, New York and London, 1973. . It is shown that Ž1. with a fixed y w Žor with fixed x Ž1 .y w. is equivalent to the inequality Ž2. with fixed y w Žor with fixed x w, respectively.. Then these results are applied to study some conditional inequalities for deviation means