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Sharp bounds for the Toader mean in terms of arithmetic and geometric means

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

For \(a,b>0\) with \(a\ne b\), the Toader mean of a and b is defined by

$$\begin{aligned} T\left( a,b\right) =\frac{2}{\pi }\int _{0}^{\pi /2}\sqrt{a^{2}\cos ^{2}t+b^{2}\sin ^{2}t}dt. \end{aligned}$$

In this paper, we prove that the double inequality

$$\begin{aligned} \left( \frac{3}{2}\left( \frac{a+b}{2}\right) ^{q}-\frac{1}{2}\left( \sqrt{ab }\right) ^{q}\right) ^{1/q}<T\left( a,b\right) <\left( \frac{3}{2}\left( \frac{a+b}{2}\right) ^{p}-\frac{1}{2}\left( \sqrt{ab}\right) ^{p}\right) ^{1/p} \end{aligned}$$

holds if and only if \(0<p\le 3/2\) and \(q\ge \ln \left( 3/2\right) /\ln \left( 4/\pi \right) \). This gives new sharp lower and upper bounds for the Toader mean, and improves several known results.

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Authors and Affiliations

  1. Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, 071003, Baoding, People’s Republic of China

    Zhen-Hang Yang

  2. Zhejiang Electric Power Company Research Institute, 310014, Hangzhou, People’s Republic of China

    Zhen-Hang Yang

  3. Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, 071003, Baoding, People’s Republic of China

    Jing-Feng Tian

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  1. Zhen-Hang Yang
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This work was supported by the Fundamental Research Funds for the Central Universities under Grant 2015ZD29 and Grant 13ZD19.

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Yang, ZH., Tian, JF. Sharp bounds for the Toader mean in terms of arithmetic and geometric means. RACSAM 115, 99 (2021). https://doi.org/10.1007/s13398-021-01040-4

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