Bounds for the Arithmetic Mean in Terms of the Neuman-Sándor and Other Bivariate Means

Abstract

We present the largest valuesα1,α2, andα3and the smallest valuesβ1,β2, andβ3such that the double inequalitiesα1M(a,b)+(1-α1)H(a,b)<A(a,b)<β1M(a,b)+ (1-β1)H(a,b),α2M(a,b)+(1-α2) H-(a,b) < A(a,b)<β2M(a,b)+(1-β2)H-(a,b), andα3M(a,b)+(1-α3)He(a,b)< A(a,b)<β3M (a,b)+(1-β3)He(a,b)hold for alla,b>0witha≠b, whereM(a,b),A(a,b),He(a,b),H(a,b)andH-(a,b)denote the Neuman-Sándor, arithmetic, Heronian, harmonic, and harmonic root-square means ofaandb, respectively.