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Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

In: Mathematica Slovaca, vol. 70, no. 5
Hong-Hu Chu - Tie-Hong Zhao - Yu-Ming Chu

Details:

Year, pages: 2020, 1097 - 1112
Keywords:
Toader mean, complete elliptic integral, arithmetic mean, quadratic mean, contraharmonic me
About article:
In the article, we present the best possible parameters $α1, β1, α2, β2\in \mathbb{R}$ and $α3, β3\in [1/2, 1]$ such that the double inequalities \begin{equation*} \aligned α1C(a, b)+(1-α1)A(a, b) &3(a, b)<β1C(a, b)+(1-β1)A(a, b), α2C(a, b)+(1-α2)Q(a, b) &3(a, b)<β2C(a, b)+(1-β2)Q(a, b), C(α3; a, b) &3(a, b)3; a, b) \endaligned \end{equation*} hold for $a, b>0$ with $a\neq b$, and provide new bounds for the complete elliptic integral of the second kind, where $A(a, b)=(a+b)/2$ is the arithmetic mean, $Q(a, b)=\sqrt{(a2+b2)/2}$ is the quadratic mean, $C(a, b)=(a2+b2)/(a+b)$ is the contra-harmonic mean, $C(p; a, b)=C[pa+(1-p)b, pb+(1-p)a]$ is the one-parameter contra-harmonic mean and $T3(a,b)=\Big(((2) / (π))\int0π/2\sqrt{a3\cos2θ+b3\sin2θ}\ddθ\Big)2/3$ is the Toader mean of order 3.
How to cite:
ISO 690:
Chu, H., Zhao, T., Chu, Y. 2020. Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means. In Mathematica Slovaca, vol. 70, no.5, pp. 1097-1112. 0139-9918. DOI: https://doi.org/DOI: 10.1515/ms-2017-0417

APA:
Chu, H., Zhao, T., Chu, Y. (2020). Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means. Mathematica Slovaca, 70(5), 1097-1112. 0139-9918. DOI: https://doi.org/DOI: 10.1515/ms-2017-0417
About edition:
Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava
Published: 27. 9. 2020