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Optimal combinations bounds of root-square and arithmetic means for Toader mean

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Abstract

We find the greatest value α 1 and α 2, and the least values β 1 and β 2, such that the double inequalities α 1 S(a,b) + (1 − α 1) A(a,b) < T(a,b) < β 1 S(a,b) + (1 − β 1) A(a,b) and \(S^{\alpha_{2}}(a,b)A^{1-\alpha_{2}}(a,b)< T(a,b)< S^{\beta_{2}}(a,b)A^{1-\beta_{2}}(a,b)\) hold for all a,b > 0 with a ≠ b. As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions. Here, S(a,b) = [(a 2 + b 2)/2]1/2, A(a,b) = (a + b)/2, and \(T(a,b)=\frac{2}{\pi}\int\limits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}{\rm d}\theta\) denote the root-square, arithmetic, and Toader means of two positive numbers a and b, respectively.

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Authors and Affiliations

  1. Department of Mathematics and Computing Science, Hunan City University, Yiyang, 413000, China

    YU-MING CHU

  2. Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, China

    MIAO-KUN WANG

  3. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    SONG-LIANG QIU

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  1. YU-MING CHU
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  2. MIAO-KUN WANG
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  3. SONG-LIANG QIU
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Correspondence to YU-MING CHU.

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CHU, YM., WANG, MK. & QIU, SL. Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc Math Sci 122, 41–51 (2012). https://doi.org/10.1007/s12044-012-0062-y

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  • DOI: https://doi.org/10.1007/s12044-012-0062-y

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