Optimal Lehmer Mean Bounds for the Toader Mean

Abstract

We find the greatest value p and least value q such that the double inequality Lp(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.