Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

Abstract

We present the best possible parameters α=α(r) and β=β(r) such that the double inequality Mα(a,b)0 with a≠b, where Mp(a, b)=[(ap+bp)/2]1/p (p≠0) and M0(a, b)=ab and Hr(a, b)=2[ra+(1-r)b][rb+(1-r)a]/(a+b) are the power and one-parameter harmonic means of a and b, respectively.