, Volume 61, Issue 3-4, pp 223-229

Optimal Lehmer Mean Bounds for the Toader Mean

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Abstract

We find the greatest value p and least value q such that the double inequality L p (a, b) < T(a, b) < L q (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 L p (a, b) = (a p+1 + b p+1)/(a p  + b p ) denote the Toader and p-th Lehmer means of two positive numbers a and b, respectively.

This work is supported by the National Natural Science Foundation of China (11071069), the Natural Science Foundation of Zhejiang Province (Y7080106) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (T200924).