The optimal convex combination bounds for Seiffert's mean (Q535536)

The authors prove the following optimal bounds for the Seiffert mean \(P(a,b)=(a-b)/[2\arcsin ((a-b)/(a+b))]\) by convex combinations of contraharmonic mean \(C(a,b)=(a^{2}+b^{2})/(a+b)\) and geometric mean \(G(a,b)= \sqrt{ab}\), respectively, harmonic mean \(H(a,b)=2ab/(a+b)\). 1) The double inequality \(\alpha _{1}C(a,b)+(1-\alpha _{1})G(a,b)<P(a,b)<\beta _{1}C(a,b)+(1-\beta _{1})G(a,b)\) holds for all \( a,b>0\) with \(a\neq b\) if and only if \(\alpha _{1}\leq 2/9\) and \(\beta _{1}\geq 1/\pi\). 2) The double inequality \(\alpha _{2}C(a,b)+(1-\alpha _{2})H(a,b)<P(a,b)<\beta _{2}C(a,b)+(1-\beta _{2})H(a,b)\) holds for all \( a,b>0\) with \(a\neq b\) if and only if \(\alpha _{2}\leq 1/\pi \) and \(\beta _{2}\geq 5/12\).