... into two cases.7This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon.Sharp bounds for ... downloaded,printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Journal of Inequalities and Applications go to http://www.journalofinequalitiesandapplications.com/authors/instructions/ For ... (1 −β)H(a, b)holds for all a, b > 0 with a = b. For fixed a, b > 0 with a = b, let x ∈ [1/2, 1] andf(x) = S(xa + (1 −x)b, xb + (1 − x)a ).Then it is not difficult to verify that f(x)...