Sharp bounds for Neuman means in terms of one-parameter family of bivariate means

Abstract

Abstract We present the best possible parameters p 1 , p 2 , p 3 , p 4 , q 1 , q 2 , q 3 , q 4 ∈ [ 0 , 1 ] such that the double inequalities G p 1 ( a , b ) < S H A ( a , b ) < G q 1 ( a , b ) , Q p 2 ( a , b ) < S C A ( a , b ) < Q q 2 ( a , b ) , H p 3 ( a , b ) < S A H ( a , b ) < H q 3 ( a , b ) , C p 4 ( a , b ) < S A C ( a , b ) < C q 4 ( a , b ) hold for all a , b > 0 with a ≠ b , where S H A , S C A , S A H , S A C are the Neuman means, and G p , Q p , H p , C p are the one-parameter means. MSC:26E60.