A New Class of Operator Monotone Functions via Operator Means

TL;DR

This paper introduces a new class of operator monotone functions derived from convex inequalities, linking them to weighted logarithmic means and extending inequalities among various operator means.

Contribution

It develops a novel class of operator monotone functions via convex inequalities and explores their relationships with weighted operator means, including extensions of key inequalities.

Findings

Established a new class of operator monotone functions related to weighted logarithmic means.

Extended inequalities among operator means, including arithmetic, geometric, and identric means.

Connected the new class to operator connections via the Hermite--Hadamard inequality.

Abstract

In this paper, we obtain a new class of functions, which is developed via the Hermite--Hadamard inequality for convex functions. The well-known one-one correspondence between the class of operator monotone functions and operator connections declares that the obtained class represents the weighted logarithmic means. We shall also consider weighted identric mean and some relationships between various operator means. Among many things, we extended the weighted arithmetic--geometric operator mean inequality as $A\#_{t}B\leq A\ell_t B\leq \frac{1}{2}(A\#_{t}B + A\nabla_{t} B)\le A\nabla_tB$ and $A\#_{t}B\leq A\mathcal{I}_t B\leq A\nabla_{t} B$ involving the considered operator means.