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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
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