Optimal evaluations for the S\'{a}ndor-Yang mean by power mean

TL;DR

This paper establishes precise bounds for the Sandor-Yang mean using power means, identifying exact parameter ranges where the inequalities hold, thus advancing the understanding of mean inequalities.

Contribution

It provides the first exact characterization of the parameter bounds for inequalities involving the Sandor-Yang mean and power means.

Findings

The inequality holds if and only if p ≤ 1.2351 and q ≥ 4/3.

The paper proves the bounds are tight and optimal.

It introduces a new approach to mean inequalities using these parameters.

Abstract

In this paper, we prove that the double inequality $M_{p}(a,b)<B(a,b)<M_{q}(a,b)% holds for all $a, b>0$ with $a\neq b$ if and only if $p\leq 4\log 2/(4+2\log 2-\pi)=1.2351\cdots$ and $q\geq 4/3$, where $% M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$ $(r\neq 0)$ and $M_{0}(a,b)=\sqrt{ab}$ is the $r$th power mean, $B(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$ is the S\'{a}% ndor-Yang mean, $A(a,b)=(a+b)/2$, $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ and $% T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$.