Sharp power mean bounds for the combination of Seiffert and geometric means (Q1957554)

Summary: We answer the question: for \(\alpha \in (0,1)\), what are the greatest value \(p\) and the least value \(q\) such that the double inequality \(M_{p}(a,b)<P^\alpha(a,b)G^{1-\alpha}(a,b)<M_q(a,b)\) holds for all \(a,b>0\) with \(a\neq b\). Here, \(M_{p}(a,b)\), \(P(a,b)\), and \(G(a,b)\) denote the power of order \(p\), Seiffert, and geometric means of two positive numbers \(a\) and \(b\), respectively.