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Zbl 1210.26021
Chu, Yu-Ming; Wang, Miao-Kun; Qiu, Ye-Fang
An optimal double inequality between power-type Heron and Seiffert means.
(English)
[J] J. Inequal. Appl. 2010, Article ID 146945, 11 p. (2010). ISSN 1029-242X/e

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: $$H_k(a,b)=\cases \bigl((a^k+(a\,b)^{k/2}+b^k)/3\bigr)^{1/k}, &k\neq 0,\\ \sqrt{a\,b}, &k=0, \endcases$$ and $$T(a,b)=\cases (a-b)/2\arctan\bigl((a-b)/(a+b)\bigr), &a\neq b,\\ a,&a=b, \endcases$$ 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.
[Roman Wituła (Gliwice)]
MSC 2000:
*26D15 Inequalities for sums, series and integrals of real functions
26E60 Means

Keywords: power-type Heron mean; Seiffert mean

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