Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means

Abstract

We present the best possible parametersα1,α2,β1,β2∈Randα3,β3∈(1/2,1)such that the double inequalitiesα1A(a,b)+(1-α1)C(a,b)<NQA(a,b)<β1A(a,b)+(1-β1)C(a,b),Aα2(a,b)C1-α2(a,b)<NQA(a,b)<Aβ2(a,b)C1-β2(a,b),andC[α3a+(1-α3)b,α3b+(1-α3)a]<NQA(a,b)<C[β3a+(1-β3)b,β3b+(1-β3)a]hold for alla,b>0witha≠band give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here,N(a,b),A(a,b),Q(a,b), andC(a,b)are, respectively, the Neuman, arithmetic, quadratic, and centroidal means ofaandb, andNQA(a,b)=N[Q(a,b),A(a,b)].