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The optimal convex combination bounds for Seiffert’s mean. (English) Zbl 1221.26037
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 $$\beta _{2}\geq 5/12$$.

##### MSC:
 2.6e+61 Means
Full Text:
##### References:
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