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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
##### Keywords:
power-type Heron mean; Seiffert mean
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##### References:
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