The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

For p is an element of R, the power mean M-p(a, b) of order p, logarithmic mean L(a, b), and arithmetic mean A(a, b) of two positive real values a and b are defined by M-p(a, b) = ((a(p) + b(p))/2)(1/p), for p not equal 0 and M-p(a, b) = root ab, for p = 0, L(a, b) = (b - a)/(log b - log a), for a not equal b and L(a, b) = a, for a = b and A(a, b) = (a + b)/2, respectively. In this paper, we answer the question: for alpha is an element of (0, 1), what are the greatest value p and the least value q, such that the double inequality M-p(a, b) <= alpha A(a, b) + (1 - alpha)L(a, b) <= M-q(a, b) holds for all a, b > 0?