A Best Possible Double Inequality for Power Mean

Abstract

We answer the question: for any p,q∈ℝ with p≠q and p≠-q, what are the greatest value λ=λ(p,q) and the least value μ=μ(p,q), such that the double inequality Mλ(a,b)0 with a≠b? Where Mp(a,b) is the pth power mean of two positive numbers a and b.