# zbMATH — the first resource for mathematics

Optimal one-parameter mean bounds for the convex combination of arithmetic and logarithmic means. (English) Zbl 1333.26035
Summary: We find the greatest value $$p_1 = p_1(\alpha )$$ and the least value $$p_2 = p_2(\alpha )$$ such that the double inequality $$J_{p_1}(a,b) < \alpha A(a,b)+(1-\alpha )L(a,b)< J_{p_2}(a,b)$$ holds for any $$\alpha \in (0,1)$$ and all $$a,b>0$$ with $$a\neq b$$. Here, $$A(a,b)$$, $$L(a,b)$$ and $$J_p(a,b)$$ denote the arithmetic, logarithmic and $$p$$th one-parameter means of two positive numbers $$a$$ and $$b$$, respectively.

##### MSC:
 2.6e+61 Means
##### Keywords:
one-parameter mean; arithmetic mean; logarithmic mean
Full Text: