Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means

Abstract

We prove that the double inequalities Iα1(a,b)Q1-α1(a,b)0 with a≠b if and only if α1≥1/2, β1≤log[2log(1+2)]/(1-log2), α2≥5/7, and β2≤log[2log(1+2)], where I(a,b), M(a,b), Q(a,b), and C(a,b) are the identric, Neuman-Sándor, quadratic, and contraharmonic means of a and b, respectively.