Research article

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

  • Received: 31 March 2020 Accepted: 13 May 2020 Published: 21 May 2020
  • MSC : 33C05, 33E05

  • 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.

    Citation: Tie-Hong Zhao, Miao-Kun Wang, Yu-Ming Chu. A sharp double inequality involving generalized complete elliptic integral of the first kind[J]. AIMS Mathematics, 2020, 5(5): 4512-4528. doi: 10.3934/math.2020290

    Related Papers:

  • 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.


    加载中


    [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. doi: 10.1016/j.jmaa.2011.10.063
    [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. doi: 10.1016/j.jmaa.2018.03.005
    [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. doi: 10.1216/RMJ-2016-46-2-679
    [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. doi: 10.1186/s13660-017-1594-6
    [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. doi: 10.1016/j.jmaa.2019.02.018
    [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. doi: 10.1007/s13398-019-00732-2
    [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. doi: 10.1016/j.jmaa.2011.06.039
    [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. doi: 10.1016/j.camwa.2011.12.038
    [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. doi: 10.1080/10652469.2011.609482
    [14] Y. M. Chu, M. Adil Khan, T. Ali, et al. Inequalities for α-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 1-12. doi: 10.1186/s13660-016-1272-0
    [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. doi: 10.1186/s13660-016-1272-0
    [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. doi: 10.1007/s11139-017-9888-3
    [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. doi: 10.1007/s11139-018-0130-8
    [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. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1186/s13660-019-2265-6
    [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. doi: 10.3934/math.2020171
    [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. doi: 10.1016/j.aml.2010.12.044
    [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. doi: 10.1007/s00025-010-0090-9
    [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. doi: 10.1016/j.jmaa.2012.05.083
    [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. doi: 10.1016/j.jat.2012.03.011
    [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. doi: 10.1007/s12044-012-0062-y
    [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. doi: 10.1186/s13660-016-1272-0
    [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. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1002/mana.200810197
    [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. doi: 10.1016/j.jmva.2011.08.004
    [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. doi: 10.1515/math-2017-0121
    [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. doi: 10.1186/s13660-017-1594-6
    [43] S. Zaheer Ullah, M. Adil Khan, Y. M. Chu, A note on generalized convex functions, J. Inequal. Appl., 2019 (2019), 1-10. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1007/s13398-018-0615-8
    [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. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1186/s13660-019-1955-4
    [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. doi: 10.1007/s13398-019-00643-2
    [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. doi: 10.1007/s40315-020-00298-w
    [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. doi: 10.1186/s13662-019-2438-0
    [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. doi: 10.1002/mma.6066
    [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. doi: 10.1186/s13662-019-2438-0
    [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. doi: 10.1007/s13398-019-00732-2
    [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. doi: 10.1186/s13660-017-1594-6
    [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. doi: 10.2996/kmj/1458651700
    [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. doi: 10.1016/j.aml.2011.09.038
    [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. doi: 10.1216/RMJ-2014-44-5-1661
    [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. doi: 10.1007/s13398-019-00732-2
    [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. doi: 10.3934/math.2020229
    [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. doi: 10.1186/s13660-019-2265-6
    [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. doi: 10.1137/0521029
    [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. doi: 10.1186/s13660-016-1272-0
    [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. doi: 10.1186/s13660-016-1272-0
    [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. doi: 10.1186/s13660-019-2265-6
    [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. doi: 10.1186/s13660-017-1594-6
    [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.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(6305) PDF downloads(470) Cited by(187)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog