Around Strongly Operator Convex Functions

TL;DR

This paper explores properties and inequalities of strongly operator convex functions across various intervals, establishing subadditivity, relations with operator monotone functions, and bounds on differences of operator values.

Contribution

It introduces new subadditivity inequalities for strongly operator convex functions and connects these with operator monotone functions and Kwong functions.

Findings

Established subadditivity inequalities for strongly operator convex functions.

Derived bounds for differences of operator functions based on convexity properties.

Connected strongly operator convex functions with operator monotone and Kwong functions.

Abstract

In this paper, we obtain the subadditivity inequality of strongly operator convex functions on $(0, \infty)$ and $(-\infty,0)$. Applying the properties of operator convex functions, we deduce the subadditivity property of operator monotone functions on $(0, \infty)$. We give new operator inequalities involving strongly operator convex functions on an interval and the weighted operator means. We also investigate relations between strongly operator convex functions and Kwong functions on $(0, \infty)$. Moreover, we study the strongly operator convex functions on $(a, \infty)$ with $a>-\infty$ and also on left half line $(-\infty,b)$ with $b< \infty$. We show that any non-constant strongly operator convex function on $(a, \infty)$ is strictly operator decreasing, and any non-constant strongly operator convex function on $(-\infty,b)$ is strictly operator monotone. Consequently, if $g$ is a strongly operator convex function on $(a, \infty)$ or $(-\infty,b)$, we estimate lower bounds of $|g(A)-g(B)|$ whenever $A-B>0$.