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 ${\Lambda }_{f,g}(a,b)$ of mean values where $f,g$ are continuously differentiable convex functions satisfying the relation $f"(t)=tg"(t),t\in {\mathbb{R}}^{+}$. Then we asked for a characterization of $f,g$ such that the inequalities $H(a,b)\le {\Lambda }_{f,g}(a,b)\le A(a,b)$ or $L(a,b)\le {\Lambda }_{f,g}(a,b)\le I(a,b)$ hold for each positive $a,b$, where $H,A,L,I$ are the harmonic, arithmetic, logarithmic and identric means, respectively. For a subclass of $\Lambda$ with $g"(t)=t^s,s\in \mathbb{R}$, this problem is thoroughly solved.