AIMS Mathematics, 2020, 5(5): 4512-4528. doi: 10.3934/math.2020290

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

A sharp double inequality involving generalized complete elliptic integral of the first kind

Tie-Hong Zhao, Miao-Kun Wang, Yu-Ming Chu

1 Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, P. R. China
2 Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China
3 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China

Received: , Accepted: , Published:

Abstract    Full Text(HTML)    Figure/Table    Related pages

In the article, we establish a sharp double inequality involving the ratio of generalized complete elliptic integrals of the first kind, which is the improvement and generalization of some previously known results.
  Figure/Table
  Supplementary
  Article Metrics

References

1. M. K. Wang, Y. M. Chu, S. L. Qiu, et al. Convexity of the complete elliptic integrals of the first kind with respect to Hölder means, J. Math. Anal. Appl., 388 (2012), 1141-1146.    

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

3. Z. H. Yang, W. M. Qian, Y. M. Chu, Monotonicity properties and bounds involving the complete elliptic integrals of the first kind, Math. Inequal. Appl., 21 (2018), 1185-1199.

4. Z. H. Yang, W. M. Qian, W. Zhang, et al. Notes on the complete elliptic integral of the first kind, Math. Inequal. Appl., 23 (2020), 77-93.

5. M. K. Wang, Y. M. Chu, Y. P. Jiang, Ramanujan's cubic transformation inequalities for zerobalanced hypergeometric functions, Rocky Mountain J. Math., 46 (2016), 679-691.    

6. M. K. Wang, Y. M. Chu, Refinements of transformation inequalities for zero-balanced hypergeometric functions, Acta Math. Sci., 37B (2017), 607-622.

7. T. H. Zhao, M. K. Wang, W. Zhang, et al. Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl., 2018 (2018), 1-15.    

8. S. L. Qiu, X. Y. Ma, Y. M. Chu, Sharp Landen transformation inequalities for hypergeometric functions, with applications, J. Math. Anal. Appl., 474 (2019), 1306-1337.    

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

10. T. H. Zhao, L. Shi, Y. M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, RACSAM, 114 (2020), 1-14.    

11. M. K. Wang, S. L. Qiu, Y. M. Chu, et al. Generalized Hersch-Pfluger distortion function and complete elliptic integrals, J. Math. Anal. Appl., 385 (2012), 221-229.    

12. Y. M. Chu, M. K. Wang, S. L. Qiu, et al. Bounds for complete elliptic integrals of the second kind with applications, Comput. Math. Appl., 63 (2012), 1177-1184.    

13. Y. M. Chu, Y. F. Qiu, M. K. Wang, Hölder mean inequalities for the complete elliptic integrals, Integral Transforms Spec. Funct., 23 (2012), 521-527.    

14. Y. M. Chu, M. Adil Khan, T. Ali, et al. Inequalities for α-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 1-12.    

15. Z. H. Yang, W. M. Qian, Y. M. Chu, et al. Monotonicity rule for the quotient of two functions and its application, J. Inequal. Appl., 2017 (2017), 1-13.    

16. M. K. Wang, Y. M. Li, Y. M. Chu, Inequalities and infinite product formula for Ramanujan generalized modular equation function, Ramanujan J., 46 (2018), 189-200.    

17. M. K. Wang, Y. M. Chu, W. Zhang, Precise estimates for the solution of Ramanujan's generalized modular equation, Ramanujan J., 49 (2019), 653-668.    

18. S. H. Wu, Y. M. Chu, Schur m-power convexity of generalized geometric Bonferroni mean involving three parameters, J. Inequal. Appl., 2019 (2019), 1-11.    

19. M. A. Latif, S. Rashid, S. S. Dragomir, et al. Hermite-Hadamard type inequalities for coordinated convex and qausi-convex functions and their applications, J. Inequal. Appl., 2019 (2019), 1-33.    

20. I. Abbas Baloch, Y. M. Chu, Petrović-type inequalities for harmonic h-convex functions, J. Funct. Space., 2020 (2020), 1-7.

21. X. M. Hu, J. F. Tian, Y. M. Chu, et al. On Cauchy-Schwarz inequality for N-tuple diamond-alpha integral, J. Inequal. Appl., 2020 (2020), 1-15.    

22. S. Rashid, M. A. Noor, K. I. Noor, et al. Ostrowski type inequalities in the sense of generalized K-fractional integral operator for exponentially convex functions, AIMS Mathematics, 5 (2020), 2629-2645.    

23. M. K. Wang, Y. M. Chu, Y. F. Qiu, et al. An optimal power mean inequality for the complete elliptic integrals, Appl. Math. Lett., 24 (2011), 887-890.    

24. G. D. Wang, X. H. Zhang, Y. M. Chu, A power mean inequality for the Grötzsch ring function, Math. Inequal. Appl., 14 (2011), 833-837.

25. Y. M. Chu, M. K. Wang, Inequalities between arithmetic-geometric, Gini, and Toader means, Abstr. Appl. Anal., 2012 (2012), 1-11.

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

27. Y. M. Chu, M. K. Wang, Y. P. Jiang, et al. Concavity of the complete elliptic integrals of the second kind with respect to Hölder means, J. Math. Anal. Appl., 395 (2012), 637-642.    

28. M. K. Wang, Y. M. Chu, S. L. Qiu, et al. Bounds for the perimeter of an ellipse, J. Approx. Theory, 164 (2012), 928-937.    

29. Y. M. Chu, M. K. Wang, S. L. Qiu, Optimal combinations bounds of root-square and arithmetic means for Toader mean, P. Indian Acad. Sci. Math. Sci., 122 (2012), 41-51.    

30. W. M. Qian, Y. M. Chu, Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters, J. Inequal. Appl., 2017 (2017), 1-10.    

31. W. M. Qian, X. H. Zhang, Y. M. Chu, Sharp bounds for the Toader-Qi mean in terms of harmonic and geometric means, J. Math. Inequal., 11 (2017), 121-127.

32. M. K. Wang, S. L. Qiu, Y. M. Chu, Infinite series formula for Hübner upper bound function with applications to Hersch-Pfluger distortion function, Math. Inequal. Appl., 21 (2018), 629-648.

33. T. H. Zhao, B. C. Zhou, M. K. Wang, et al. On approximating the quasi-arithmetic mean, J. Inequal. Appl., 2019 (2019), 1-12.    

34. J. L. Wang, W. M. Qian, Z. Y. He, et al. On approximating the Toader mean by other bivariate means, J. Funct. Space., 2019 (2019), 1-7.

35. W. M. Qian, Y. Y. Yang, H. W. Zhang, et al. Optimal two-parameter geometric and arithmetic mean bounds for the Sándor-Yang mean, J. Inequal. Appl., 2019 (2019), 1-12.    

36. Y. M. Chu, G. D. Wang, X. H. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetric function, Math. Nachr., 284 (2011), 653-663.    

37. Y. M. Chu, B. Y. Long, Sharp inequalities between means, Math. Inequal. Appl., 14 (2011), 647-655.

38. Y. M. Chu, W. F. Xia, X. H. Zhang, The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications, J. Multivariate Anal., 105 (2012), 412-421.    

39. M. Adil Khan, Y. M. Chu, T. U. Khan, et al. Some new inequalities of Hermite-Hadamard type for s-convex functions with applications, Open Math., 15 (2017), 1414-1430.    

40. Y. Q. Song, M. Adil Khan, S. Zaheer Ullah, et al. Integral inequalities involving strongly convex functions, J. Funct. Space., 2018 (2018), 1-8.

41. M. Adil Khan, Y. M. Chu, A. Kashuri, et al. Conformable fractional integrals versions of HermiteHadamard inequalities and their generalizations, J. Funct. Space., 2018 (2018), 1-9.

42. H. Z. Xu, Y. M. Chu, W. M. Qian, Sharp bounds for the Sándor-Yang means in terms of arithmetic and contra-harmonic means, J. Inequal. Appl., 2018 (2018), 1-13.    

43. S. Zaheer Ullah, M. Adil Khan, Y. M. Chu, A note on generalized convex functions, J. Inequal. Appl., 2019 (2019), 1-10.    

44. M. K. Wang, H. H. Chu, Y. M. Chu, Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals, J. Math. Anal. Appl., 480 (2019), 1-9.

45. M. Adil Khan, M. Hanif, Z. A. Khan, et al. Association of Jensen's inequality for s-convex function with Csiszár divergence, J. Inequal. Appl., 2019 (2019), 1-14.    

46. M. Adil Khan, S. Zaheer Ullah, Y. M. Chu, The concept of coordinate strongly convex functions and related inequalities, RACSAM, 113 (2019), 2235-2251.    

47. S. Zaheer Ullah, M. Adil Khan, Z. A. Khan, et al. Integral majorization type inequalities for the functions in the sense of strong convexity, J. Funct. Space., 2019 (2019), 1-11.

48. S. Zaheer Ullah, M. Adil Khan, Y. M. Chu, Majorization theorems for strongly convex functions, J. Inequal. Appl., 2019 (2019), 1-13.    

49. M. Adil Khan, S. H. Wu, H. Ullah, et al. Discrete majorization type inequalities for convex functions on rectangles, J. Inequal. Appl., 2019 (2019), 1-18.    

50. Y. Khurshid, M. Adil Khan, Y. M. Chu, Conformable integral inequalities of the HermiteHadamard type in terms of GG- and GA-convexities, J. Funct. Space., 2019 (2019), 1-8.

51. Y. Khurshid, M. Adil Khan, Y. M. Chu, et al. Hermite-Hadamard-Fejér inequalities for conformable fractional integrals via preinvex functions, J. Funct. Space., 2019 (2019), 1-9.

52. W. M. Qian, Z. Y. He, H. W. Zhang, et al. Sharp bounds for Neuman means in terms of twoparameter contraharmonic and arithmetic mean, J. Inequal. Appl., 2019 (2019), 1-13.    

53. W. M. Qian, H. Z. Xu, Y. M. Chu, Improvements of bounds for the Sándor-Yang means, J. Inequal. Appl., 2019 (2019), 1-8.    

54. X. H. He, W. M. Qian, H. Z. Xu, et al. Sharp power mean bounds for two Sándor-Yang means, RACSAM, 113 (2019), 2627-2638.    

55. W. M. Qian, W. Zhang, Y. M. Chu, Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means, Miskolc Math. Notes, 20 (2019), 1157-1166.

56. M. K. Wang, Z. Y. He, Y. M. Chu, Sharp power mean inequalities for the generalized elliptic integral of the first kind, Comput. Meth. Funct. Th., 20 (2020), 111-124.    

57. M. Adil Khan, N. Mohammad, E. R. Nwaeze, et al. Quantum Hermite-Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020 (2020), 1-20.    

58. S. Khan, M. Adil Khan, Y. M. Chu, Converses of the Jensen inequality derived from the Green functions with applications in information theory, Math. Method. Appl. Sci., 43 (2020), 2577-2587.    

59. A. Iqbal, M. Adil Khan, S. Ullah, et al. Some new Hermite-Hadamard-type inequalities associated with conformable fractional integrals and their applications, J. Funct. Space., 2020 (2020), 1-18.

60. S. Rafeeq, H. Kalsoom, S. Hussain, et al. Delay dynamic double integral inequalities on time scales with applications, Adv. Differ. Equ., 2020 (2020), 1-32.    

61. B. Wang, C. L. Luo, S. H. Li, et al. Sharp one-parameter geometric and quadratic means bounds for the Sándor-Yang means, RACSAM, 114 (2020), 1-10.    

62. M. K. Wang, W. Zhang, Y. M. Chu, Monotonicity, convexity and inequalities involving the generalized elliptic integrals, Acta Math. Sci., 39B (2019), 1440-1450.

63. T. R. Huang, S. Y. Tan, X. Y. Ma, et al. Monotonicity properties and bounds for the complete p-elliptic integrals, J. Inequal. Appl., 2018 (2018), 1-11.    

64. M. K. Wang, M. Y. Hong, Y. F. Xu, et al. Inequalities for generalized trigonometric and hyperbolic functions with one parameter, J. Math. Inequal., 14 (2020), 1-21.

65. S. Takeuchi, A new form of the generalized complete elliptic integrals, Kodai Math. J., 39 (2016), 202-226.    

66. Y. F. Qiu, M. K. Wang, Y. M. Chu, et al. Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean, J. Math. Inequal., 5 (2011), 301-306.

67. M. K. Wang, Z. K. Wang, Y. M. Chu, An optimal double inequality between geometric and identric means, Appl. Math. Lett., 25 (2012), 471-475.    

68. G. D. Wang, X. H. Zhang, Y. M. Chu, A power mean inequality involving the complete elliptic integrals, Rocky Mountain J. Math., 44 (2014), 1661-1667.    

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

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

71. W. M. Qian, Z. Y. He, Y. M. Chu, Approximation for the complete elliptic integral of the first kind, RACSAM, 114 (2020), 1-12.    

72. S. Rashid, M. A. Noor, K. I. Noor, et al. Hermite-Hadamrad type inequalities for the class of convex functions on time scale, Mathematics, 7 (2019), 1-20.

73. S. Rashid, F. Jarad, M. A. Noor, et al. Inequalities by means of generalized proportional fractional integral operators with respect another function, Mathematics, 7 (2019), 1-18.

74. S. Rashid, R. Ashraf, M. A. Noor, et al. New weighted generalizations for differentiable exponentially convex mappings with application, AIMS Mathematics, 5 (2020), 3525-3546.    

75. S. Khan, M. Adil Khan, Y. M. Chu, New converses of Jensen inequality via Green functions with applications, RACSAM, 114 (2020), 114.

76. M. U. Awan, S. Talib, Y. M. Chu, et al. Some new refinements of Hermite-Hadamard-type inequalities involving Ψk-Riemann-Liouville fractional integrals and applications, Math. Probl. Eng., 2020 (2020), 1-10.

77. M. U. Awan, N. Akhtar, S. Iftikhar, et al. Hermite-Hadamard type inequalities for n-polynomial harmonically convex functions, J. Inequal. Appl., 2020 (2020), 1-12.    

78. S. Rashid, F. Jarad, Y. M. Chu, A note on reverse Minkowski inequality via generalized proportional fractional integral operator with respect to another function, Math. Probl. Eng., 2020 (2020), 1-12.

79. G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Functional inequalities for complete elliptic integrals and ratios, SIAM J. Math. Anal., 21 (1990), 536-549.    

80. H. Alzer, K. Richards, Inequalities for the ratio of complete elliptic integrals, P. Am. Math. Soc., 145 (2017), 1661-1670.

81. L. Yin, L. G. Huang, Y. L. Wang, et al. An inequality for generalized complete elliptic integral, J. Inequal. Appl., 2017 (2017), 1-6.    

82. T. H. Zhao, Y. M. Chu, H. Wang, Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal., 2011 (2011), 1-13.

83. Z. H. Yang, W. M. Qian, Y. M. Chu, et al. On rational bounds for the gamma function, J. Inequal. Appl., 2017 (2017), 1-17.    

84. G. J. Hai, T. H. Zhao, Monotonicity properties and bounds involving the two-parameter generalized Grötzsch ring function, J. Inequal. Appl., 2020 (2020), 1-17.    

85. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1992.

86. T. R. Huang, B. W. Han, X. Y. Ma, et al. Optimal bounds for the generalized Euler-Mascheroni constant, J. Inequal. Appl., 2018 (2018), 1-9.    

87. S. Rashid, F. Jarad, H. Kalsoom, et al. On Pólya-Szegö and Ćebyšev type inequalities via generalized k-fractional integrals, Adv. Differ. Equ., 2020 (2020), 125.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved