## Optimal inequalities between Seiffert’s mean and power means.(English)Zbl 1049.26006

For the Seiffert mean $$P(x,y):=(x-y)/[4\arctan (\sqrt{x/y})-\pi ]$$, the author proves that the evaluation $$A_{p}\leq P\leq A_{q}$$ holds if and only if $$0<p\leq \log 2/\log \pi$$ and $$q\geq 2/3$$, where $$A_{p}$$ is the usual power mean defined by $$A_{p}(s,y):=[(x^{p}+y^{p})/2]^{1/p}.$$

### MSC:

 26D07 Inequalities involving other types of functions 26E60 Means

### Keywords:

power means; Seiffert mean; inequalities