Sharp bounds by the power mean for the generalized Heronian mean

* Correspondence: [email protected] Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China Full list of author information is available at the end of the article Abstract In this article, we answer the question: For p, ω Î 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 Mr1 (a, b) < Hp,ω (a, b) < Mr2 (a, b) 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.