Research article

Convexity and inequalities related to extended beta and confluent hypergeometric functions

  • Received: 17 July 2019 Accepted: 05 September 2019 Published: 23 September 2019
  • MSC : Primary 33B15; Secondary 26D15, 33B99

  • In the paper, the authors establish the logarithmic convexity and some inequalities for the extended beta function and, by using these inequalities for the extended beta function, find the logarithmic convexity and the monotonicity for the extended confluent hypergeometric function.

    Citation: Feng Qi, Kottakkaran Sooppy Nisar, Gauhar Rahman. Convexity and inequalities related to extended beta and confluent hypergeometric functions[J]. AIMS Mathematics, 2019, 4(5): 1499-1507. doi: 10.3934/math.2019.5.1499

    Related Papers:

  • In the paper, the authors establish the logarithmic convexity and some inequalities for the extended beta function and, by using these inequalities for the extended beta function, find the logarithmic convexity and the monotonicity for the extended confluent hypergeometric function.


    加载中


    [1] R. P. Agarwal, N. Elezović, and J. Pečarić, On some inequalities for beta and gamma functions via some classical inequalities, J. Inequal. Appl., 2005 (2005), 593-613.
    [2] P. Agarwal, M. Jleli, and F. Qi, Extended Weyl fractional integrals and their inequalities, ArXiv: 1705.03131, 2017. Available from: https://arxiv.org/abs/1705.03131.
    [3] P. Agarwal, F. Qi, M. Chand, et al. Certain integrals involving the generalized hypergeometric function and the Laguerre polynomials, J. Comput. Appl. Math., 313 (2017), 307-317. doi: 10.1016/j.cam.2016.09.034
    [4] M. Biernacki and J. Krzyż, On the monotonity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska. Sect. A, 9 (1955), 135-147.
    [5] S. I. Butt, J. Pečarić, and A. U. Rehman, Exponential convexity of Petrović and related functional, J. Inequal. Appl., 2011 (2011), 89.
    [6] M. A. Chaudhry, A. Qadir, M. Rafique, et al. Extension of Euler's beta function, J. Comput. Appl. Math., 78 (1997), 19-32. doi: 10.1016/S0377-0427(96)00102-1
    [7] M. A. Chaudhry, A. Qadir, H. M. Srivastava, et al. Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589-602.
    [8] S. S. Dragomir, R. P. Agarwal and N. S. Barnett, Inequalities for Beta and Gamma functions via some classical and new integral inequalities, J. Inequal. Appl., 5 (2000), 103-165.
    [9] D. Karp and S. M. Sitnik, Log-convexity and log-concavity of hypergeometric-like functions, J. Math. Anal. Appl., 364 (2010), 384-394. doi: 10.1016/j.jmaa.2009.10.057
    [10] P. Kumar, S. P. Singh and S. S. Dragomir, Some inequalities involving beta and gamma functions, Nonlinear Anal. Forum, 6 (2001), 143-150.
    [11] K. S. Miller and S. G. Samko, A note on the complete monotonicity of the generalized MittagLeffler function, Real Analysis Exchange, 23 (1997), 753-756.
    [12] S. R. Mondal, Inequalities of extended beta and extended hypergeometric functions, J. Inequal. Appl., 2017 (2017), 10.
    [13] K. S. Nisar and F. Qi, On solutions of fractional kinetic equations involving the generalized kBessel function, Note di Matematica, 37 (2018), 11-20.
    [14] K. S. Nisar, F. Qi, G. Rahman, et al. Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function, J. Inequal. Appl., 2018 (2018), 135.
    [15] H. Pollard, The completely monotonic character of the Mittag-Leffler function Ea(-x), B. Am. Math. Soc., 54 (1948), 1115-1116.
    [16] F. Qi, Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities, Filomat, 27 (2013), 601-604. doi: 10.2298/FIL1304601Q
    [17] F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequal. Appl., 2019 (2019), 36.
    [18] F. Qi, A. Akkurt and H. Yildirim, Catalan numbers, k-gamma and k-beta functions, and parametric integrals, J. Comput. Anal. Appl., 25 (2018), 1036-1042.
    [19] F. Qi, R. Bhukya and V. Akavaram, Inequalities of the Grünbaum type for completely monotonic functions, Adv. Appl. Math. Sci., 17 (2018), 331-339.
    [20] F. Qi, R. Bhukya and V. Akavaram, Some inequalities of the Turán type for confluent hypergeometric functions of the second kind, Stud. Univ. Babeş-Bolyai Math., 64 (2019), 63-70. doi: 10.24193/subbmath.2019.1.06
    [21] F. Qi, L. H. Cui and S. L. Xu, Some inequalities constructed by Tchebysheff's integral inequality, Math. Inequal. Appl., 2 (1999), 517-528.
    [22] F. Qi and W. H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions, J. Appl. Anal. Comput., 5 (2015), 626-634.
    [23] F. Qi and W. H. Li, Integral representations and properties of some functions involving the logarithmic function, Filomat, 30 (2016), 1659-1674. doi: 10.2298/FIL1607659Q
    [24] F. Qi and A. Q. Liu, Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math., 361 (2019), 366-371. doi: 10.1016/j.cam.2019.05.001
    [25] F. Qi and K. S. Nisar, Some integral transforms of the generalized k-Mittag-Leffler function, Publ. Inst. Math. (Beograd) (N.S.), 104 (2019), in press.
    [26] F. Qi, G. Rahman and K. S. Nisar, Convexity and inequalities related to extended beta and confluent hypergeometric functions, HAL archives, 2018. Available from:https://hal.archives-ouvertes.fr/hal-01703900.
    [27] S. L. Qiu, X. Y. Ma and 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
    [28] E. D. Rainville, Special Functions, Macmillan, New York, 1960.
    [29] W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill Book Co., New York, 1987.
    [30] M. Shadab, S. Jabee and J. Choi, An extension of beta function and its application, Far East Journal of Mathematical Sciences (FJMS), 103 (2018), 235-251. doi: 10.17654/MS103010235
    [31] J. F. Tian and M. H. Ha, Properties of generalized sharp Hölder's inequalities, J. Math. Inequal., 11 (2017), 511-525.
    [32] M. K. Wang, H. H. Chu and Y. M. Chu, Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals, J. Math. Anal. Appl., 480 (2019), 123388.
    [33] M.-K. Wang, Y.-M. Chu and Y.-P. Jiang, Ramanujan's cubic transformation inequalities for zerobalanced hypergeometric functions, Rocky MT J. Math., 46 (2016), 679-691. doi: 10.1216/RMJ-2016-46-2-679
    [34] M.-K. Wang, Y. M. Chu and W. Zhang, Monotonicity and inequalities involving zero-balanced hypergeometric function, Math. Inequal. Appl., 22 (2019), 601-617.
    [35] M. K. Wang, Y. M. Chu and 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
    [36] Z. H. Yang, W. M. Qian, Y. M. Chu, et al. On rational bounds for the gamma function, J. Inequal. Appl., 2017 (2017), 210.
  • Reader Comments
  • © 2019 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(3566) PDF downloads(423) Cited by(5)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog