Slavko Simic Mathematical Institute SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia Correspondence should be addressed to Slavko Simic; [email protected] Received 25 June 2012; Revised 9 December 2012; Accepted 16 December 2012 Academic Editor: Mowaffaq Hajja Copyright © 2013 Slavko Simic.This is an open access article distributed under the Creative Commons Attribution License, which p...

Abstract. Let p ∈ R , M be a bivariate mean, and Mp be defined by Mp(a,b) = M1/p(ap,bp) (p = 0) and M0(a,b) = limp→0 Mp(a,b) . In this paper, we prove that the sharp inequalities L2(a,b) < P(a,b) < NS1/2(a,b) < He(a,b) < A2/3(a,b) < I(a,b) < Z1/3(a,b) < Y1/2(a,b) hold for all a,b > 0 with a = b , where L(a,b) = (a− b)/(loga − logb) , P(a,b) = (a− b)/[2arcsin((a−b)/(a+b))] , NS(a,b) = (a−b)/[2ar...

and Applied Analysis 3 Theorem 1.1. If p, q ∈ 0, 1/2 , then the double inequality H ( pa ( 1 − pb, pb 1 − pa < I a, b < H ( qa ( 1 − qb, qb 1 − qa 1.8 holds for all a, b > 0 with a/ b if and only if p ≤ 1 − √ 1 − 2/e /2 and q ≥ 6 − √6 /12. 2. Proof of Theorem 1.1 Proof of Theorem 1.1. Let λ 6 − √6 /12 and μ 1 − √ 1 − 2/e /2. Then from the monotonicity of the function f x H xa 1 − x b, xb 1 − x ...