# zbMATH — the first resource for mathematics

Optimal one-parameter mean bounds for the convex combination of arithmetic and geometric means. (English) Zbl 1276.26063
Summary: In this paper, we answer the question: What are the greatest value $$p = p(\alpha)$$ and least value $$q = q(\alpha)$$ such that the double inequality $J_p(a, b) < \alpha A(a, b) + (1 - \alpha)G(a, b) < J_q(a, b)$ holds for any $$\alpha \in (0, 1)$$ and all $$a, b > 0$$ with $$a \neq b$$? Here, $$A(a, b) = \frac{a + b}{2}$$, $$G(a, b) = \sqrt{ab}$$ and $$J_p(a, b)$$ denote the arithmetic, geometric and $$p$$-th one-parameter means of $$a$$ and $$b$$, respectively.

##### MSC:
 26E60 Means 26D20 Other analytical inequalities
##### Keywords:
one-parameter mean; arithmetic mean; geometric mean
Full Text: