Sharp Geometric Mean Bounds for Neuman Means

Abstract EN

We find the best possible constants α1,α2,β1,β2∈[0,1/2] and α3,α4,β3,β4∈[1/2,1] such that the double inequalities G(α1a+(1-α1)b,α1b + (1-α1)a)0 with a≠b, where G, A, and Q are, respectively, the geometric, arithmetic, and quadratic means and NAG, NGA, NQA, and NAQ are the Neuman means.