Around Strongly Operator Convex 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 $A-B>0$.

15 pages