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
[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. |