Sharp bounds for Seiffert mean in terms of weighted power means of arithmetic mean and geometric mean

Abstract

For a,b > 0 with a = b , let P = (a− b)/(4arctana/b−π) , A = (a+ b)/2 , G = √ 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 3 A p1 + 3 G p1 )1/p1 < P < ( 2 3 A p2 + 3 G p2 )1/p2 , where p1 = 4/5 and p2 = logπ/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 . Mathematics subject classification (2010): Primary 26E60, 26D05; secondary 33B10.