An optimal double inequality between power-type Heron and Seiffert means (Q623621)

Purpose of this paper is to present the optimal upper and lower power-type Heron mean bounds for the Seiffert mean \(T(a,b)\). For \(k\in[0;+\infty),\) the power-type Heron mean \(H_k(a,b)\) and the Seiffert mean \(T(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by: \[ H_k(a,b)=\begin{cases} \bigl((a^k+(a\,b)^{k/2}+b^k)/3\bigr)^{1/k}, &k\neq 0,\\ \sqrt{a\,b}, &k=0, \end{cases} \] and \[ T(a,b)=\begin{cases} (a-b)/2\arctan\bigl((a-b)/(a+b)\bigr), &a\neq b,\\ a,&a=b, \end{cases} \] respectively. It is proved that for all \(a,b>0\), with \(a\neq b\), one has: \[ H_{\log 3/ \log(\pi/2)}(a,b)<T(a,b)<H_{5/2}(a,b) \] and both \(H\)'s are the best possible lower and upper power-type Heron mean bounds for the Seiffert mean \(T(a,b),\) respectively.