Sharp Generalized Seiffert Mean Bounds for Toader Mean - Yu-Ming Chu; Miao-Kun Wang; Song-Liang Qiu; Ye-Fang Qiu (Abstract and Applied Analysis)
For p∈[0,1], the generalized Seiffert mean of two positive numbers a and b is defined by Sp(a,b)=p(a-b)/arctan[2p(a-b)/(a+b)],  0<p≤1,  a≠b;  (a+b)/2,  p=0,  a≠b;  a,  a=b. In this paper, we find the greatest value α and least value β such that the double inequality Sα(a,b)<T(a,b)<Sβ(a,b) holds for all a,b>0 with a≠b, and give new bounds for the complete elliptic integrals of the second kind. Here, T(a,b)=(2/π)∫0π/2a2cos⁡2θ+b2sin⁡2θdθ denotes the Toader mean of two positive numbers a and b.