SHARP BOUNDS FOR SEIFFERT MEAN IN TERMS OF WEIGHTED POWER MEANS OF ARITHMETIC MEAN AND GEOMETRIC MEAN

For a, b > 0 with a not equal b, let P = (a - b)/(4arctan root a/b - pi), A = (a + b)/2, G = root ab denote the Seiffert mean, arithmetic mean, geometric mean of a and b, respectively. In this paper, we present new sharp bounds for Seiffert P in terms of weighted power means of arithmetic mean A and geometric mean G: (2/3A(p1) + 1/3G(p1))(1/p1) < P < (2/3A(p2) + 1/3G(p2))(1/p2), where p(1) = 4/5 and p(2) = log pi/2(3/2) are the best possible constants. Moreover, our sharp bounds for P are compared with other known ones, which yields a chain of inequalities involving Seiffert mean P.