Results in Mathematics

, Volume 61, Issue 3, pp 223–229

Optimal Lehmer Mean Bounds for the Toader Mean


DOI: 10.1007/s00025-010-0090-9

Cite this article as:
Chu, YM. & Wang, MK. Results. Math. (2012) 61: 223. doi:10.1007/s00025-010-0090-9


We find the greatest value p and least value q such that the double inequality Lp(a, b) < T(a, b) < Lq(a, b) holds for all a, b > 0 with a ≠ b, and give a new upper bound for the complete elliptic integral of the second kind. Here \({T(a,b)=\frac{2}{\pi}\int\nolimits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}d\theta}\) and Lp(a, b) = (ap+1 + bp+1)/(ap + bp) denote the Toader and p-th Lehmer means of two positive numbers a and b, respectively.

Mathematics Subject Classification (2010)

Primary 26E60 


Lehmer mean Toader mean power mean 
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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsHuzhou Teachers CollegeHuzhouPeople’s Republic of China

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