Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

Abstract

We present the best possible parametersα=α(r)andβ=β(r)such that the double inequalityMα(a,b)<Hr(a,b)<Mβ(a,b)holds for allr∈(0, 1/2)anda, b>0witha≠b, whereMp(a, b)=[(ap+bp)/2]1/p (p≠0)andM0(a, b)=abandHr(a, b)=2[ra+(1-r)b][rb+(1-r)a]/(a+b)are the power and one-parameter harmonic means ofaandb, respectively.