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Sharp bounds for Seiffert mean in terms of contraharmonic mean. (English) Zbl 1231.26034
Summary: We find the greatest value $$\alpha$$ and the least value $$\beta$$ in $$(1/2, 1)$$ such that the double inequality $$C(\alpha a + (1 - \alpha) b, \alpha b + (1 - \alpha)a) < T(a, b) < C(\beta a + (1 - \beta)b, \beta b + (1 - \beta)a)$$ holds for all $$a, b > 0$$ with $$a \neq b$$. Here, $$T(a, b) = (a - b)/[2 \arctan ((a - b)/(a + b))]$$ and $$C(a, b) = (a^2 + b^2)/(a + b)$$ are the Seiffert and contraharmonic means of $$a$$ and $$b$$, respectively.

MSC:
 26E60 Means 26D07 Inequalities involving other types of functions
Keywords:
convex combination
Full Text:
References:
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