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Liu, Hong; Meng, Xiang-Ju

The optimal convex combination bounds for Seiffert’s mean. (English) Zbl 1221.26037

J. Inequal. Appl. 2011, Article ID 686834, 9 p. (2011).
The authors prove the following optimal bounds for the Seiffert mean \(P(a,b)=(a-b)/[2\arcsin ((a-b)/(a+b))]\) by convex combinations of contraharmonic mean \(C(a,b)=(a^{2}+b^{2})/(a+b)\) and geometric mean \(G(a,b)= \sqrt{ab}\), respectively, harmonic mean \(H(a,b)=2ab/(a+b)\).
1) The double inequality \(\alpha _{1}C(a,b)+(1-\alpha _{1})G(a,b)<P(a,b)<\beta _{1}C(a,b)+(1-\beta _{1})G(a,b)\) holds for all \( a,b>0\) with \(a\neq b\) if and only if \(\alpha _{1}\leq 2/9\) and \(\beta _{1}\geq 1/\pi\).
2) The double inequality \(\alpha _{2}C(a,b)+(1-\alpha _{2})H(a,b)<P(a,b)<\beta _{2}C(a,b)+(1-\beta _{2})H(a,b)\) holds for all \( a,b>0\) with \(a\neq b\) if and only if \(\alpha _{2}\leq 1/\pi \) and \(\beta _{2}\geq 5/12\).
Reviewer: Gheorghe Toader (Cluj-Napoca)

Cited in 10 Documents

MSC:

26E60 Means

Keywords:

Seiffert’s mean; contraharmonic mean; harmonic mean; geometric mean
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\textit{H. Liu} and \textit{X.-J. Meng}, J. Inequal. Appl. 2011, Article ID 686834, 9 p. (2011; Zbl 1221.26037)
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