On some intermediate mean values

TL;DR

This paper investigates conditions for mean inequalities involving Jensen functionals, introduces a new class of means defined by convex functions, and characterizes when these means relate to classical means like harmonic, arithmetic, logarithmic, and identric.

Contribution

It introduces a new class of mean values based on convex functions and characterizes their inequalities with classical means, extending the understanding of mean relations.

Findings

Characterization of mean inequalities involving Jensen functionals.

Definition of a new class of mean values $\Lambda_{f,g}$.

Complete solution for a subclass where $g''(t)=t^s$.

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 $f"(t)=t g"(t), t\in \Bbb R^+$. 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 $g"(t)=t^s, s\in \Bbb R$, this problem is thoroughly solved.