Abstract
For any α∈0,1, we answer the questions: what are the greatest values p and λ and the least values q and μ, such that the inequalities Lpa,b<Iαa,bL1-αa,b<Lqa,b and Lλa,b<αIa,b+1-αLa,b<Lμa,b hold for all a,b>0 with a≠b? Here, Ia,b, La,b, and Lpa,b denote the identric, logarithmic, and pth Lehmer means of two positive numbers a and b, respectively.
Highlights
B > 0 with a ≠ b, the logarithmic mean L(a, b) and identric mean I(a, b) are defined by L (a, b) = log a a − − b log b (1) I (a, b) 1 e ( bb aa 1/(b−a) )
For any α ∈ (0, 1), we answer the questions: what are the greatest values p and λ and the least values q and μ, such that the inequalities Lp(a, b) < Iα(a, b)L1−α(a, b) < Lq(a, b) and Lλ(a, b) < αI(a, b) + (1 − α)L(a, b) < Lμ(a, b) hold for all a, b > 0 with a ≠ b? Here, I(a, b), L(a, b), and Lp(a, b) denote the identric, logarithmic, and pth Lehmer means of two positive numbers a and b, respectively
The purpose of this paper is to present the best possible upper and lower Lehmer mean bounds of the product Iα(a, b)L1−α(a, b) and the sum αI(a, b) + (1 − α)L(a, b) for any α ∈ (0, 1) and all a, b > 0 with a ≠ b
Summary
I(a, b), L(a, b), and Lp(a, b) denote the identric, logarithmic, and pth Lehmer means of two positive numbers a and b, respectively. In [14, 17, 20], inequalities between L, I, and the classical arithmetic-geometric mean of Gauss are proved.
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