On some intermediate mean values

Abstract

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class Λf,g (a, b) of mean values where f, g are continuously differentiable convex functions satisfying the relation f ″ (t) = t g ″ (t), t +. Then we asked for a characterization of f, g such that the inequalities H (a, b) ≤ Λf,g (a, b) ≤ A (a, b) or L (a, b) ≤ Λf,g (a, b) ≤ I (a, b) hold for each positive a, b, where H, A, L, I are the harmonic, arithmetic, logarithmic, and identric means, respectively. For a subclass of Λ with g ″ (t) = ts, s, this problem is thoroughly solved.