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

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$if and only if$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}$is the S\'{a}$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))]$.