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Optimal convex combination bounds of Seiffert and geometric means for the arithmetic mean. (English) Zbl 1225.26061
Summary: We find the greatest value $$\alpha$$ and the least value $$\beta$$ such that the double inequality $$\alpha T(a,b) + (1 - \alpha )G(a,b) < A(a,b) < \beta T(a,b) + (1 - \beta )G(a,b)$$ holds for all $$a,b > 0$$ with $$a \neq b$$. Here $$T (a,b), G(a,b)$$, and $$A(a,b)$$ denote the Seiffert, geometric, and arithmetic means of two positive numbers $$a$$ and $$b$$, respectively.

##### MSC:
 2.6e+61 Means
##### Keywords:
Seiffert mean; geometric mean; arithmetic mean
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