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Sharp power mean bounds for two Sándor–Yang 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

In the article, we prove that the double inequalities

$$\begin{aligned} M_{\alpha }(a,b)< & {} Q(a,b)e^{G(a,b)/U(a,b)-1}<M_{\beta }(a,b), \\ M_{\lambda }(a,b)< & {} G(a,b)e^{Q(a,b)/V(a,b)-1}<M_{\mu }(a,b) \end{aligned}$$

hold for all \(a, b>0\) with \(a\ne b\) if and only if \(\alpha \le 2\log 2/(2+\log 2)=0.5147\cdots \), \(\beta \ge 2/3\), \(\lambda \le 2\log 2/(2-\log 2)=1.0607\cdots \) and \(\mu \ge 4/3\), where \(M_{p}(a,b)=[(a^{p}+b^{p})/2]^{1/p}\)\((p\ne 0)\), \(M_{0}(a,b)=G(a,b)=\sqrt{ab}\), \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\), \(U(a,b)=(a-b)/[\sqrt{2\breve{}}\arctan ((a-b)/\sqrt{2ab})]\) and \(V(a,b)=(a-b)/[\sqrt{2}\sinh ^{-1}((a-b)/\sqrt{2ab})]\) are respectively the pth power, geometric, quadratic, first Yang and second Yang means, and \(\sinh ^{-1}(x)\) is the inverse hyperbolic sine function.

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Acknowledgements

Funding was provided by National Natural Science Foundation of China (Grant no. 61673169) and Natural Science Foundation of Huzhou City (Grant no. 2018YZ07).

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Authors and Affiliations

  1. Office of Academic Affairs, Quzhou Broadcast and TV University, Quzhou, 324000, China

    Xiao-Hong He

  2. School of Continuing Education, Huzhou Vocational and Technical College, Huzhou, 313000, Zhejiang, China

    Wei-Mao Qian

  3. School of Economics and Management, Wenzhou Broadcast and TV University, Wenzhou, 325000, China

    Hui-Zuo Xu

  4. Department of Mathematics, Huzhou University, Huzhou, 313000, China

    Yu-Ming Chu

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This work was supported by the Natural Science Foundation of China (Grant no. 61673169) and the Natural Science Foundation of Huzhou City (Grant no. 2018YZ07).

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He, XH., Qian, WM., Xu, HZ. et al. Sharp power mean bounds for two Sándor–Yang means. RACSAM 113, 2627–2638 (2019). https://doi.org/10.1007/s13398-019-00643-2

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