Sharp two-parameter bounds for the identric mean

摘要

For (tin [0,1/2]) and (sge 1), we consider the two-parameter family of means $$ Q_{t,s}(a,b)=G^{s}igl(ta+(1-t)b,(1-t)a+tbigr)A^{1-s}(a,b), $$ where A and G denote the arithmetic and geometric means. Sharp bounds for the identric mean in terms of (Q_{t,s}) are obtained. Our results generalize and extend bounds due to Chu et al. (Abstr. Appl. Anal. 2011:657935, 2011) and to Wang et al. (Appl. Math. Lett. 25:471a??475, 2012).KeywordsArithmetic Mean??Geometric Mean??Harmonic Mean??Identric Mean??MSC26E60??26D07??1 IntroductionThe study of inequalities involving means has become very popular in recent years because of their applications in various kinds of areas of mathematics. Finding sharp bounds for inequalities is an important task in order to have more accurate results in the aforementioned areas.