Abstract
Let \({\mathcal {K}}\left( r\right) \) be the complete elliptic integrals of the first kind and \(\text{ arth }r\) denote the inverse hyperbolic tangent function. We prove that the inequality
holds for \(r\in \left( 0,1\right) \) with the best constants \(\lambda =3/4\) and \(q=1/10\). This improves some known results and gives a positive answer for a conjecture on the best upper bound for the Gaussian arithmetic–geometric mean in terms of logarithmic and arithmetic means.
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Dedicated to the 60th anniversary of Zhejiang Electric Power Company Research Institute.
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This work was supported by the Fundamental Research Funds for the Central Universities under Grant 2015ZD29, Grant 13ZD19, and Grant MS117.
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Yang, ZH., Tian, JF. & Zhu, YR. A sharp lower bound for the complete elliptic integrals of the first kind. RACSAM 115, 8 (2021). https://doi.org/10.1007/s13398-020-00949-6
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DOI: https://doi.org/10.1007/s13398-020-00949-6
Keywords
- Arithmetic–geometric mean
- Logarithmic mean
- Complete elliptic integrals of the first kind
- Inverse hyperbolic tangent function
- NP type power series
- Inequality