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On the absolute monotonicity of generalized elliptic integral of the first kind

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

Abstract

It was proved by Wang et al. (Appl Anal Discrete Math 14:255–271, 2020) that twice derivative of the function \(x\mapsto {{\,\mathrm{\textit{K}}\,}}(\sqrt{x})-\log (1+4/\sqrt{1-x})\) is absolutely monotonic on (0, 1). This will be, in this paper, extended to the generalized elliptic integral of the first kind, more precisely, we will study the absolutely monotonic properties of the function \(x\mapsto {{\,\mathrm{\textit{K}}\,}}_a(\sqrt{x})-\log \Big (1+c/\sqrt{1-x}\Big )\) on (0, 1) for \(a\in (0,1/2]\) and \(c\in (0,\infty )\), where \({{\,\mathrm{\textit{K}}\,}}_{a}\) is the generalized elliptic integral of the first kind.

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References

  1. Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Conformal Invariants, Inequalities, and Quasiconformal Maps. John Wiley & Sons, New York (1997)

    MATH  Google Scholar 

  2. Anderson, G.D., Qiu, S.-L., Vamanamurthy, M.K., Vuorinen, M.: Generalized elliptic integrals and modular equations. Pacific. J. Math. 192(1), 1–37 (2000)

    Article  MathSciNet  Google Scholar 

  3. Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, New York (1971)

    Book  MATH  Google Scholar 

  4. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462(2), 1714–1726 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Huang, T.-R., Qiu, S.-L., Ma, X.-Y.: Monotonicity properties and inequalities for the generalized elliptic integral of the first kind. J. Math. Anal. Appl. 469(1), 95–116 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Richards, K.C.: A note on inequalities for the ratio of zero-balanced hypergeometric functions. Proc. Am. Math. Soc. Ser. B 6, 15–20 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yang, Z.-H., Tian, J.-F.: Sharp inequalities for the generalized elliptic integrals of the first kind. Ramanujan J. 48(1), 91–116 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Alzer, H., Richards, K.C.: A concavity property of the complete elliptic integral of the first kind. Integral Transforms Spec Funct. 31(9), 758–768 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hai, G.-J., Zhao, T.-H.: Monotonicity properties and bounds involving the twoparameter generalized Grötzsch ring function. J. Inequal. Appl. 2020(17), Article 66 (2020)

  10. Qian, W.-M., He, Z.-H., Chu, Y.-M.: Approximation for the complete elliptic integral of the first kind. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114(2), Paper No. 57 (2020)

  11. Zhao, T.-H., He, Z.-Y., Chu, Y.-M.: Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals. Comput. Methods Funct. Theory 21(3), 413–426 (2020)

    Article  MATH  Google Scholar 

  12. Zhao, T.-H., He, Z.-Y., Chu, Y.-M.: On some refinements for inequalities involving zero-balanced hypergeometric function. AIMS Math. 6(5), 6479–6495 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Richards, K.C., Smith, J.N.: A concavity property of generalized complete elliptic integrals. Integral Transforms Spec Funct. 32(3), 240–252 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhao, T.-H., Wang, M.-K., Chu, Y.-M.: Monotonicity and convexity involving generalized elliptic integral of the first kind. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 115(2), Paper No. 46, 13 pp (2021)

  15. Chen, Y. J., Zhao, T. H.: On the Convexity and Concavity of Generalized Complete Elliptic Integral of the First Kind. Results in Math. 77(6), Paper No. 215, 20 pp (2022)

  16. Zhao, T.-H., Wang, M.-K., Hai G.-J., Chu, Y.-M.: Landen inequalities for Gaussian hypergeometric function. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116(1), Paper No. 53, 23 pp (2022)

  17. Bao, Q., Wang, M. K., Zhang, Y. A.: One answer to an open problem on the monotonicity of Gaussian hypergeometric functions with respect to parameters. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116(3), Paper No. 115, 10 pp (2022)

  18. Qian, W.-M., Wang, M.-K., Xu, H.-Z., Chu, Y.-M.: Approximations for the complete elliptic integral of the second kind. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 115(2), Paper No. 88, 11 pp (2021)

  19. Yin, L., Lin, X. L.: Monotonicity and inequalities related to the generalized inverse lemniscate functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116(1), Paper No. 52, 13 pp (2022)

  20. Zhu, L.: A new upper bound for the complete elliptic integral of the first kind. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 117(3), Paper No. 125 (2023)

  21. Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.: Functional inequalities for hypergeometric functions and complete elliptic integrals. SIAM J. Math. Anal. 23(2), 512–524 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Qiu, S.-L., Vamanamurthy, M.K., Vuorinen, M.: Some inequalities for the growth of elliptic integrals. SIAM J. Math. Anal. 29(5), 1224–1237 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yang, Z.H., Tian, J.-F.: Convexity and monotonicity for elliptic integrals of the first kind and applications. Appl. Anal. Discrete Math. 13(1), 240–260 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, M.-K., Chu, H.-H., Li, Y.-M., Chu, Y.-M.: Answers to three conjectures on the convexity of three functions involving complete elliptic integrals of the first kind. Appl. Anal. Discrete Math. 14(1), 255–271 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhao, T.-H., Wang, M.-K., Chu, Y.-M.: Monotonicity and convexity involving generalized elliptic integral of the first kind. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 115(2), Article No. 6, 13 pp (2021)

  26. Chen, Y.-J., Zhao, T.-H.: On the monotonicity for generalized elliptic integral of the first kind. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 116(2), Paper No. 77, 21 pp (2022)

  27. Ying, W.-T.: A new approach to prove a two-sided inequality involving Wallis’s formula. J. Taizhou Univ. 30(3), 1–4 (2008). ((in Chinese))

    Google Scholar 

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Funding

This work was supported by the National Natural Science Foundation of China (11971142) and the Natural Science Foundation of Zhejiang Province (LY19A010012).

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  1. School of mathematics, Hangzhou Normal University, Hangzhou, 311121, People’s Republic of China

    Yajun Chen, Jiahui Wu & Tiehong Zhao

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  1. Yajun Chen
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  2. Jiahui Wu
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  3. Tiehong Zhao
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Correspondence to Tiehong Zhao.

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Chen, Y., Wu, J. & Zhao, T. On the absolute monotonicity of generalized elliptic integral of the first kind. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 143 (2023). https://doi.org/10.1007/s13398-023-01472-0

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