On bounds of logarithmic mean and mean inequality chain
Shigeru Furuichi, Mehdi Eghbali Amlashi
TL;DR
The paper investigates bounds for the logarithmic mean $L(a,b)$, combining scalar and operator perspectives. It leverages a self-improvement approach on the fundamental chain $G \le G^{2/3}A^{1/3} \le L \le B_{1/3} \le \tfrac{2}{3}G+\tfrac{1}{3}A$ to derive refined scalar bounds and matrix-mean inequalities, and shows the optimality of the $2/3$ exponents/coefficients. The analysis extends to matrix means $K_r$, $H_s$, $B_p$, and related means, establishing monotonicity in their parameters for unitarily invariant norms and identifying both positive results and counterexamples (e.g., $K_{2/3} \npreceq L$). Finally, a new mean-inequality chain is proposed by introducing $\rho=\sqrt{\dfrac{2A}{A+G}}$, producing a tightened hierarchy $m \le \rho m \le H \le \rho H \le G \le \rho G \le L \le \rho L \le A \le \rho A \le M$, together with connections to known bounds such as $L \le \tfrac{2}{3}G+\tfrac{1}{3}A \le \tfrac{A+G}{2}$.
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.