Optimal evaluations for the Sándor-Yang mean by power mean

Abstract

In this paper, we prove that the double inequality $M_p(a,b)<B(a,b)<M_q(a,b)$a, b>0$with$a\neq b$ifandonlyif$p\leq 4\log 2/(4+2\log 2-π)=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}$isthe$r$thpowermean,$B(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$istheS\acute{a}$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))]$.

9 pages