Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means

Abstract

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