Functions with convex means

Abstract

Introduction* Let / be Lebesgue summable on [0, ], a > 0. The function F, defined on [0, a] by F x {x) = l/x[ m f(t)dt, F(0) =/(0) Jo is called the mean of the function /. Inductively, we may define the Nth mean of / on [0, a] by F N {x) = l/x\* F^ifydt, F N (0)=f(0), Jo provided, of course, that F Nt is summable on [0, ]. Some questions involving the mean of certain classes of functions have been examined in Beckenbach [1] and Bruckner and Ostrow [2].