Abstract
Abstract In this article, we answer the question: For p, ω ∈ ℝ with ω > 0 and p(ω - 2) ≠ 0, what are the greatest value r 1 = r 1(p, ω) and the least value r 2 = r 2(p, ω) such that the double inequality M r 1 a , b < H p , ω a , b < M r 2 a , b holds for all a, b > 0 with a ≠ b? Here H p,ω (a, b) and M r (a, b) denote the generalized Heronian mean and r th power mean of two positive numbers a and b, respectively. 2010 Mathematics Subject Classification: 26E60.
Highlights
The bivariate means have been the subject of intensive research
Many remarkable inequalities can be found in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]
If p >0 and 0 < ω
Summary
The bivariate means have been the subject of intensive research. In particular, many remarkable inequalities can be found in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].The power mean Mr(a, b) of order r of two positive numbers a and b is defined by ⎧ ⎨ ar +br Mr (a, b) = ⎩ √2 ab, 1/r , r = 0, r = 0. (1:1)It is well-known that. The power mean Mr(a, b) of order r of two positive numbers a and b is defined by In [27], Alzer and Janous established the following sharp double inequality
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