Some families of operator norm inequalities

TL;DR

This paper investigates operator norm inequalities derived from specific classes of functions involving power and product forms, expanding the understanding of positive definite functions and their implications in operator theory.

Contribution

It introduces a new family of functions with a particular structure and explores their role in establishing operator norm inequalities, extending previous results.

Findings

Identified conditions under which the functions imply positive definiteness.

Established new operator norm inequalities for the introduced function family.

Extended the class of functions known to generate such inequalities.

Abstract

It is known that the function $f(e^x)/g(e^x)$ is positive definite for some functions $f,g$ implies the operator norm inequality related to $f,g$. We treat functions which have the following form: $f(t) = t^{(1-\sum_{i=1}^n (a_i-b_i))/2}\prod_{i=1}^n \frac{b_i(t^{a_i}-1)}{a_i(t^{b_i}-1)}$.