Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means

We answer the question: for α,β,γ∈(0,1) with α+β+γ=1, what are the greatest value p and the least value q, such that the double inequality Lp(a,b)0 with a≠b? Here Lp(a,b), A(a,b), G(a,b), and H(a,b) denote the generalized logarithmic, arithmetic, geometric, and harmonic means of two positive numbers a and b, respectively