Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means - Tie-Hong Zhao (Abstract and Applied Analysis, 2013)
We prove that the double inequalities Iα1(a,b)Q1-α1(a,b)<M(a,b)<Iβ1(a,b)Q1-β1(a,b),Iα2(a,b)C1-α2(a,b)<M(a,b)<Iβ2(a,b)C1-β2(a,b) hold for all 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.