On bounds of logarithmic mean and mean inequality chain

TL;DR

This paper investigates bounds and inequalities related to the logarithmic mean, providing operator and norm inequalities, monotonicity properties, and a new mean inequality chain, advancing the theoretical understanding of mean inequalities.

Contribution

It introduces new bounds and inequalities for the logarithmic mean, including operator and norm inequalities, and establishes a novel mean inequality chain.

Findings

Operator inequalities for the logarithmic mean

Monotonicity and optimality of the Heron mean

Ordering of various unitarily invariant norms

Abstract

An upper bound of the logarithmic mean is given by a convex combination of the arithmetic mean and the geometric mean. In addition, a lower bound of the logarithmic mean is given by a geometric bridge of the arithmetic mean and the geometric mean. In this paper, we study the bounds of the logarithmic mean. We give operator inequalities and norm inequalities for the fundamental inequalities on the logarithmic mean. We give monotonicity of the parameter for the unitarily invariant norm of the Heron mean, and give its optimality as the upper bound of the unitarily invariant norm of the logarithmic mean. We study the ordering of the unitarily invariant norms for the Heron mean, the Heinz mean, the binomial mean and the Lehmer mean. Finally, we give a new mean inequality chain as an application of the point-wise inequality.