Two optimal double inequalities between power mean and logarithmic mean

For p is an element of R the power mean M(p)(a, b) of order p, the logarithmic mean L(a, b) and the arithmetic mean A(a, b) of two positive real values a and b are defined by M(p)(a, b) = {(a(p) + b(p))(1/p), p not equal 0, root ab, p = 0 L(a, b) = {b - a/log b - log a, a not equal b, a, a = b, and A(a, b) = a+b/2, respectively. In this article, we answer the questions: What are the greatest values p and r, and the least values q and s, such that the inequalities M(p)(a, b) <= 1/3A(a, b) + 2/3L(a, b) <= M(q)(a, b) and M(r) (a, b) <= 2/3A(a, b) + 1/3L(a, b) <= M(s)(a, b) hold for all a, b > 0? (C) 2010 Elsevier Ltd. All rights reserved.