Optimal Inequalities for Generalized Logarithmic,Arithmetic, and Geometric Means

摘要

For p ∈ R, the generalized logarithmic mean L_p(a,b), arithmetic mean A(a,b), and geometric mean G{a,b) of two positive numbers a and b are defined by L_p(a,b) = a, for a = b, L_p{a,b) = [(b~(p+1)-a~(p+1))/((p + 1)(b-a))]~(1/p), for p≠0, p≠-1,and a≠b, L_p(a,b) = (1/e)(b~b/a~a)~(1/b-a), for p = 0, and a≠b, L_p(a,b) = (b - a)/(log b - log a), for p = -1, and a≠b, A(a,b) = (a + b)/2, and G(a, b) = ab~1/2, respectively. In this paper, we find the greatest value p (or least value q, resp.) such that the inequality L_p(a,b) < αA(a,b) + (1 -α)G(a,b) (or αA(a,b) + (1-α)G(a,b) < L_q(a,b), resp.) holds for α ∈ (0,l/2)(or α ∈ (1/2,1), resp.) and all a,b > 0 with a≠b.