Operator $k$-tone functions and analytic functional calculus

TL;DR

This paper characterizes operator k-tone functions, extending operator monotone and convex functions, through inequalities involving their analytic functional calculus, advancing understanding of their mathematical properties.

Contribution

It introduces a new characterization of operator k-tone functions using inequalities related to their analytic functional calculus, broadening the theoretical framework.

Findings

Operator k-tone functions are characterized via inequalities.

The paper extends the theory of operator monotone and convex functions.

Analytic functional calculus plays a key role in the characterization.

Abstract

Operator $k$-tone functions on an open interval of the real line, which are higher order extensions of operator monotone and convex functions, are characterized via certain inequalities for the real and imaginary parts of analytic functional calculus by those functions.