Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means

Abstract

We present the best possible parameters α1,α2,β1,β2∈R and α3,β3∈(1/2,1) such that the double inequalities α1A(a,b)+(1-α1)C(a,b)0 with a≠b and give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here, N(a,b), A(a,b), Q(a,b), and C(a,b) are, respectively, the Neuman, arithmetic, quadratic, and centroidal means of a and b, and NQA(a,b)=N[Q(a,b),A(a,b)].