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arXiv · 1402.5020

A double inequality for bounding Toader mean by the centroidal mean

Yun Hua·Feng Qi
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Abstract

In the paper, the authors find the best numbers $\alpha$ and $\beta$ such that $$ \overline{C}\bigl(\alpha a+(1-\alpha)b,\alpha b+(1-\alpha)a\bigr) 0$ with $a\ne b$, where $\overline{C}(a,b)={2\bigl(a^2+ab+b^2\bigr)}{3(a+b)}$ and $T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}\,d\theta$ denote respectively the centroidal mean and Toader mean of two positive numbers $a$ and $b$.

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Yun Hua, Feng Qi. 2014-02-20. A double inequality for bounding Toader mean by the centroidal mean. https://doi.org/10.1007/s12044-014-0183-6

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