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An optimal double inequality between power-type Heron and Seiffert means. (English) Zbl 1210.26021
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.

MSC:
26D15 Inequalities for sums, series and integrals
26E60 Means
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References:
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