Optimal Lehmer Mean Bounds for the Toader Mean |
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Authors: | Yu-Ming Chu Miao-Kun Wang |
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Institution: | 1. Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, Zhejiang, People’s Republic of China
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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. |
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