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A family of high order approximations of Ramanujan type for perimeter of an ellipse

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

Abstract

Inspired by Ramanujan’s unusual approximations for the perimeter of an ellipse with semiaxis 1 and \(r\in \left( 0,1\right) \), a family of approximations of Ramanujan type is constructed. We find the sharp lower and upper approximations have very high accuracy, and better than some known ones.

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References

  1. Bowman, F.: Introduction to Elliptic Functions with Applications. Dover Publications, New York (1961)

    MATH  Google Scholar 

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

    Book  Google Scholar 

  3. 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. RACSAM 114, 126 (2021)

  4. Huang, X.-F., Wang, M.-K., Shao, H., Zhao, Y.-F., Chu, Y.-M.: Monotonicity properties and bounds for the complete p-elliptic integrals. AIMS Math. 5(6), 7071–7086 (2020)

    Article  MathSciNet  Google Scholar 

  5. Zhao, T.-H., Wang, M.-K., Chu, Y.-M.: A sharp double inequality involving generalized complete elliptic integral of the first kind. AIMS Math. 5(5), 4512–4528 (2020)

    Article  MathSciNet  Google Scholar 

  6. Wang, M.-K., Chu, Y.-M., Li, Y.-M., Zhang, W.: Asymptotic expansion and bounds for complete elliptic integrals. Math. Inequal. Appl. 23(3), 821–841 (2020)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Wang, M.-K., He, Z.-Y., Chu, Y.-M.: Sharp power mean inequalities for the generalized elliptic integral of the first kind. Comput. Methods Funct. Theory 20(1), 111–124 (2020)

    Article  MathSciNet  Google Scholar 

  9. Wang, M.-K., Chu, H.-H., Chu, Y.-M.: Precise bounds for the weighted Hölder mean of the complete \(p\)-elliptic integrals. J. Math. Anal. Appl. 480(2), 123388 (2019)

    Article  MathSciNet  Google Scholar 

  10. Qiu, S.-L., Ma, X.-Y., Chu, Y.-M.: Extensions of quadratic transformation identities for hypergeometric functions. Math. Inequal. Appl. 23(4), 1391–1423 (2020)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Qian, W.-M., He, Z.-Y., Chu, Y.-M.: Approximation for the complete elliptic integral of the first kind. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114(2), 57 (2020)

  13. 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. RACSAM 115(2), 46 (2020)

  14. Yang, Z.-H., Tian, J.-F., Wang, M.-K.: A positive answer to Bhatia-Li conjecture on the monotonicity for a new mean in its parameter. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114, 126 (2020)

  15. Tian, J.-F., Yang, Z.-H.: Asymptotic expansions of Gurland’s ratio and sharp bounds for their remainders. J. Math. Anal. Appl. 493(2), 124545 (2021)

    Article  MathSciNet  Google Scholar 

  16. Chu, Y.-M., Wang, M.-K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61, 223–229 (2012)

    Article  MathSciNet  Google Scholar 

  17. Chu, Y.-M., Wang, M.-K.: Inequalities between arithmetic-geometric, Gini, and Toader means. Abstr. Appl. Anal. 2012, 11 (2012) (Art. ID 830585)

  18. Chu, H.-H., Zhao, T.-H., Chu, Y.-M.: Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means. Math. Slovaca 70(5), 1097–1112 (2020)

    Article  MathSciNet  Google Scholar 

  19. Kepler, J.: Opera Omnia. Astronomia Nova. Heyder and Zimmer, Frankfurt (1860)

    Google Scholar 

  20. Almkvist, G., Berndt, B.: Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, \(\pi \), and the Ladies Diary. Am. Math. Mon. 95, 585–608 (1988)

    MathSciNet  MATH  Google Scholar 

  21. Barnard, R.W., Pearce, K., Richards, K.C.: A monotonicity property involving \(_{3}F_{2}\) and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32, 403–419 (2000)

    Article  MathSciNet  Google Scholar 

  22. Qiu, S.-L.: The Muir mean and the complete elliptic intergral of the second kind. J. Hangzhou Inst. Electr. Eng. 20, 28–33 (2000) (in Chinese)

  23. Barnard, R.W., Pearce, K., Richards, K.C.: An inequality involving the generalized hypergeometric function and the arc length of an ellipse. SIAM J. Math. Anal. 31, 693–699 (2000)

    Article  MathSciNet  Google Scholar 

  24. Alzer, H., Qiu, S.-L.: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172, 289–312 (2004)

    Article  MathSciNet  Google Scholar 

  25. Kazi, H., Neuman, E.: Inequalities and bounds for elliptic integrals. J. Approx. Theory 146, 212–226 (2007)

    Article  MathSciNet  Google Scholar 

  26. Chu, Y.-M., Wang, M.-K., Qiu, S.-L., Jiang, Y.-P.: Bounds for complete elliptic integrals of the second kind with applications. Comput. Math. Appl. 63, 1177–1184 (2012)

    Article  MathSciNet  Google Scholar 

  27. Wang, M.-K., Chu, Y.-M., Qiu, S.-L., Jiang, Y.-P.: Bounds for the perimeter of an ellipse. J. Approx. Theory 164, 928–937 (2012)

    Article  MathSciNet  Google Scholar 

  28. Wang, M.-K., Chu, Y.-M.: Asymptotical bounds for complete elliptic integrals of the second kind. J. Math. Anal. Appl. 402, 119–126 (2013)

    Article  MathSciNet  Google Scholar 

  29. Li, W.-H., Zheng, M.-M.: Some inequalities for bounding Toader mean. J. Funct. Spaces 2013, 5 (2013) (Art. ID 394194)

  30. Yang, Z.-H., Chu, Y.-M., Zhang, W.: Accurate approximations for the complete elliptic integral of the second kind. J. Math. Anal. Appl. 438, 875–888 (2016)

    Article  MathSciNet  Google Scholar 

  31. Yang, Z.-H.: Sharp approximations for the complete elliptic integrals of the second kind by one-parameter means. J. Math. Anal. Appl. 467, 446–461 (2018)

    Article  MathSciNet  Google Scholar 

  32. Wang, J.-L., Qian, W.-M., He, Z.-Y., Chu, Y.-M.: On approximating the Toader mean by other bivariate means. J. Funct. Spaces 2019, 7 (2019) (Art. ID 6082413)

  33. Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Ramanujan, S.: Notebooks (2 volumes). Tata Institute of Fundamental Research, Bombay (1957)

    MATH  Google Scholar 

  35. Biernacki, M., Krzyz, J.: On the monotonicity of certain functionals in the theory of analytic functions. Ann. Univ. Mariae Curie Sklodowska 9, 135–147 (1955)

    MATH  Google Scholar 

  36. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl. 2017, 13 (2017) (Article 106)

  37. Yang, Z.-H., Chu, Y.-M., Wang, M.-K.: Monotonicity criterion for the quotient of power series with applications. J. Math. Anal. Appl. 428, 587–604 (2015)

    Article  MathSciNet  Google Scholar 

  38. Wang, M.-K., Chu, Y.-M., Song, Y.-Q.: Asymptotical formulas for Gaussian and generalized hypergeometric functions. Appl. Math. Comput. 276, 44–60 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Yang, Z.-H., Chu, Y.-M.: A monotonicity property involving the generalized elliptic integrals of the first kind. Math. Inequal. Appl. 20, 729–735 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the error function. Math. Inequal. Appl. 21, 469–479 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Yang, Z.-H., Zheng, S.-Z.: Monotonicity of the ratio of modified Bessel functions of the first kind with applications. J. Inequal. Appl. 2018, 57 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  43. Wang, M.-K., Chu, Y.-M., Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 22, 601–617 (2019)

    MathSciNet  MATH  Google Scholar 

  44. Yang, Z.-H., Qian, W.-M., Zhang, W., Chu, Y.-M.: Notes on the complete elliptic integral of the first kind. Math. Inequal. Appl. 23, 77–93 (2020)

    MathSciNet  MATH  Google Scholar 

  45. Yang, Z.-H.: A new way to prove L’Hospital Monotone Rules with applications (2020). arXiv:1409.6408 [math.CA]

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

    Article  MathSciNet  Google Scholar 

  47. Yang, Z.-H., Zheng, S.-Z.: Sharp bounds for the ratio of modified Bessel functions. Mediterr. J. Math. 14, 169 (2017)

    Article  MathSciNet  Google Scholar 

  48. Lindner, G.: Lexikon der gesamten Technik und ihrer Hilfswissenschaften. Deutsche Verlagsanstalt, Stuttgart (1904)

    Google Scholar 

  49. Selmer, E.S.: Bemerkninger til en ellipse-beregning av en ellipses ornkrets. Nordisk Mat. Tidskr. 23, 55–58 (1975)

    Google Scholar 

  50. Almkvist, G.: Aritmetisk-geometriska medelvlirdet och ellipsens bagllingd. Nordisk Mat. Tidskr. 25–26, 121–130 (1978)

    MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities (No. 2015ZD29) and the National Natural Science Foundation of China (No. 61672205).

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Authors and Affiliations

  1. Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, Baoding, 071003, People’s Republic of China

    Jing-Feng Tian

  2. Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, Baoding, 071003, People’s Republic of China

    Zhenhang Yang

  3. Zhejiang Society for Electric Power, Hangzhou, 310014, Zhejiang, People’s Republic of China

    Zhenhang Yang

  4. School of Science, Hebei University of Engineering, Taiji Road 19, Handan, 056038, People’s Republic of China

    Ming-Hu Ha

  5. College of Mathematics and Information Science, Hebei University, Wusi Road 180, Baoding, 071002, People’s Republic of China

    Hong-Jie Xing

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  1. Jing-Feng Tian
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  2. Zhenhang Yang
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Correspondence to Zhenhang Yang.

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Tian, JF., Yang, Z., Ha, MH. et al. A family of high order approximations of Ramanujan type for perimeter of an ellipse. RACSAM 115, 85 (2021). https://doi.org/10.1007/s13398-021-01021-7

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