Some results on strongly operator convex functions and operator monotone functions

TL;DR

This paper offers an operator inequality-based approach to studying strongly operator convex, operator monotone, and operator convex functions, providing new methods for constructing such functions and exploring their interrelations.

Contribution

It introduces an alternative operator inequality approach to analyze strongly operator convex functions and establishes a characterization linking operator monotonicity to strong operator convexity.

Findings

Operator inequalities can replace operator algebraic methods.

Characterization of operator monotone functions via strong operator convexity.

Methods for constructing new functions in the three classes.

Abstract

This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and [4], where operator algebraic semicontinuity theory or operator theory were substantially used. In this paper we provide an alternate treatment that uses only operator inequalities (or even just matrix inequalities). We also show that if t_0 is a point in the domain of a continuous function f, then f is operator monotone if and only if (f(t) - f(t_0))/(t - t_0) is strongly operator convex. Using this and previously known results, we provide some methods for constructing new functions in one of the three classes from old ones. We also include some discussion of completely monotone functions in this context and some results on the operator convexity or strong operator convexity of phi \circ f when f is operator convex or strongly operator convex.