Sharp Generalized Seiffert Mean Bounds for Toader Mean

For p is an element of [0, 1], the generalized Seiffert mean of two positive numbers a and b is defined by S-p (a, b)=p(a-b)/arctan[2p(a-b)/(a+b)], 0 < p <= 1, a not equal b; (a+b)/2, p=0, a not equal b; a, a=b. In this paper, we find the greatest value a and least value beta such that the double inequality S-a(a, b) < T (a, b) < S-beta(a, b) holds for all a, b > 0 with a not equal b, and give new bounds for the complete elliptic integrals of the second kind. Here, T(a, b)=(2/pi) integral(pi/2)(0) root a(2)cos(2)theta+b(2)sin(2)theta d theta denotes the Toader mean of two positive numbers a and b.