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Sharp bounds for Seiffert mean in terms of root mean square. (English) Zbl 1275.26051

Summary: We find the greatest value \(\alpha\) and least value \(\beta\) in \((1/2,1)\) such that the double inequality \[ S(\alpha a+(1-\alpha)b,\alpha b+(1-\alpha)a)<T(a,b)< S(\beta a+(1-\beta)b,\beta b+(1-\beta)a) \] holds for all \(a,b>0\) with \(a\neq b\). Here, \(T(a,b)=(a-b)/[2 \arctan((a-b)/(a+b))]\) and \(S(a,b)=[(a^2+b^2)/2]^{1/2}\) are the Seiffert mean and root mean square of \(a\) and \(b\), respectively.

MSC:

26E60 Means
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References:

[1] doi:10.1016/j.jco.2003.08.007 · Zbl 1073.65005 · doi:10.1016/j.jco.2003.08.007
[2] doi:10.1063/1.532988 · Zbl 1055.82507 · doi:10.1063/1.532988
[3] doi:10.1139/p80-162 · Zbl 0982.81538 · doi:10.1139/p80-162
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