Open Access Research

Sharp bounds by the power mean for the generalized Heronian mean

Yong-Min Li1, Bo-Yong Long2 and Yu-Ming Chu1*

Author Affiliations

1 Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

2 School of Mathematical Science, Anhui University, Hefei 230039, China

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Journal of Inequalities and Applications 2012, 2012:129 doi:10.1186/1029-242X-2012-129

Published: 7 June 2012

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 r1 = r1(p, ω) and the least value r2 = r2(p, ω) such that the double inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/129/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/129/mathml/M1">View MathML</a> holds for all a, b > 0 with a ≠ b? Here Hp,ω(a, b) and Mr(a, b) denote the generalized Heronian mean and rth power mean of two positive numbers a and b, respectively.

2010 Mathematics Subject Classification: 26E60.

Keywords:
generalized Heronian mean; power mean; Heronian mean.