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arXiv · 1506.07777

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

Zhen-Hang Yang·Yu-Ming Chu
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Abstract

In this paper, we prove that the double inequality $M_{p}(a,b) 0$ 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 $% T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$.

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Zhen-Hang Yang, Yu-Ming Chu. 2015-06-25. Optimal evaluations for the S\'{a}ndor-Yang mean by power mean. https://arxiv.org/abs/1506.07777

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