Mathematica Eterna
Open Access

Abstract

Optimal Inequalities for Generalized Logarithmic and Seiffert Means

Shaoqin Gao, Lingling Song, and Mengna You

For r ∈ R , the generalized logarithmic mean Lr(a, b) and Seiffert mean P(a, b) of two positive numbers a and b are defined by Lr(a, b) = a, for a = b, Lr(a, b) = [(b r − a r )/r(b− a)] 1 r−1 , for r 6= 1, r 6= 0, and a 6= b, Lr(a, b) = 1 e ( b b a a ) 1 b−a , for r = 1 and a 6= b, Lr(a, b) = (b − a)/(ln b − ln a), for r = 0 and a 6= b, and P(a, b) = (a − b)/(4 arctan p a/b − π) respectively. In this paper, we find the greatest value α and the least value β such that the inequality Lα(a, b) < P(a, b) (or P(a, b) < Lβ(a, b), resp.) holds for all a, b > 0 with a 6= b.