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Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind

  • Received: 17 May 2021 Accepted: 29 July 2021 Published: 06 August 2021
  • MSC : 30C45, 33C10

  • A useful family of integral operators and special functions plays a crucial role on the study of mathematical and applied sciences. The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk. Furthermore, we determine the order of the convexity of the families of integral operators. In order to prove main results, we use differential inequalities for the Lommel functions of the first kind together with some known properties in connection with the integral operators which we have considered in this paper. We also indicate the connections of the results presented here with those in several earlier works on the subject of our investigation. Moreover, some graphical illustrations are provided in support of the results proved in this paper.

    Citation: Ji Hyang Park, Hari Mohan Srivastava, Nak Eun Cho. Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind[J]. AIMS Mathematics, 2021, 6(10): 11380-11402. doi: 10.3934/math.2021660

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  • A useful family of integral operators and special functions plays a crucial role on the study of mathematical and applied sciences. The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk. Furthermore, we determine the order of the convexity of the families of integral operators. In order to prove main results, we use differential inequalities for the Lommel functions of the first kind together with some known properties in connection with the integral operators which we have considered in this paper. We also indicate the connections of the results presented here with those in several earlier works on the subject of our investigation. Moreover, some graphical illustrations are provided in support of the results proved in this paper.



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    [1] L. V. Ahlfors, Sufficient conditions for quasiconformal extension, Ann. Math. Stud., 79 (1974), 23-29.
    [2] H. A. Al-Kharsani, A. M. Al-Zahrani, S. S. Al-Hajri, T. K. Pogány, Univalence criteria for linear fractional differential operators associated with a generalized Bessel function, Math. Commun. 21 (2016), 171-188.
    [3] Á. Baricz, D. K. Dimitrov, H. Orhan, N. Yağmur, Radii of starlikeness of some special functions, Proc. Amer. Math. Soc., 144 (2016), 3355-3367. doi: 10.1090/proc/13120
    [4] Á. Baricz, B. A. Frasin, Univalence of integral operators involving Bessel functions, Appl. Math. Lett., 23 (2010), 371-376. doi: 10.1016/j.aml.2009.10.013
    [5] Á. Baricz, S. Koumandos, Turán type inequalities for some Lommel function of the first kind, Proc. Edinburgh Math. Soc., 59 (2016), 569-579. doi: 10.1017/S0013091515000413
    [6] Á. Baricz, R. Szász, Close-to-convexity of some special functions and their derivatives, Bull. Malays. Math. Sci. Soc., 39 (2016), 427-437. doi: 10.1007/s40840-015-0180-7
    [7] J. Becker, Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte funktionen, J. Reine Angew. Math., 255 (1972), 23-43.
    [8] J. Becker, Löwnersche Differentialgleichung und Schlichtheitskriterien, Math. Ann., 202 (1973), 321-335. doi: 10.1007/BF01433462
    [9] S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 35 (1969), 429-446.
    [10] D. Breaz, N. Breaz, Univalence conditions for certain integral operators, Stud. Univ. Babeş-Bolyai Math., 47 (2002), 9-15.
    [11] D. Breaz, N. Breaz, Univalence of an integral operator, Mathematica (Cluj), 47 (2005), 35-38.
    [12] D. Breaz, N. Breaz, H. M. Srivastava, An extension of the univalent condition for a family of integral operators, Appl. Math. Lett., 22 (2009), 41-44. doi: 10.1016/j.aml.2007.11.008
    [13] D. Breaz, S. Owa, N. Breaz, A new integral univalent operator, Acta Univ. Apulensis Math. Inform., 16 (2008), 11-16.
    [14] B. C. Carlson, D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 159 (1984), 737-745.
    [15] N. E. Cho, O. S. Kwon, H. M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators, J. Math. Anal. Appl., 292 (2004), 470-483. doi: 10.1016/j.jmaa.2003.12.026
    [16] J. H. Choi, M. Saigo, H. M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl., 276 (2002), 432-445. doi: 10.1016/S0022-247X(02)00500-0
    [17] E. Deniz, H. Orhan, H. M. Srivastava, Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions, Taiwanese J. Math., 15 (2011), 883-917.
    [18] E. Deniz, Convexity of integral operators involving generalized Bessel functions, Integral Transforms Spec. Funct., 24 (2013), 201-216. doi: 10.1080/10652469.2012.685938
    [19] A. W. Goodman, Univalent Functions, Mariner Publishing Company Incorporated, Tampa, FL, 1983.
    [20] Y. J. Kim, E. P. Merkes, On an integral of powers of a spirallike function, Kyungpook Math. J., 12 (1972), 249-252.
    [21] R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc., 16 (1965), 755-758. doi: 10.1090/S0002-9939-1965-0178131-2
    [22] J. L. Liu, The Noor integral and strongly starlike functions, J. Math. Anal. Appl., 261 (2001), 441-447. doi: 10.1006/jmaa.2001.7489
    [23] E. Lommel, Über eine mit den Bessel'schen Functionen verwandte Function, Math. Ann., 9 (1875), 425-444. doi: 10.1007/BF01443342
    [24] S. Moldoveanu, N. N. Pascu, Integral operators which preserve the univalence, Mathematica (Cluj), 32 (1990), 159-166.
    [25] K. I. Noor, On new classes of integral operators, J. Natur. Geom., 16 (1999), 71-80.
    [26] K. I. Noor, H. A. Alkhorasani, Properties of close-to-convexity preserved by some integral operators, J. Math. Anal. Appl., 112 (1985), 509-516. doi: 10.1016/0022-247X(85)90260-4
    [27] S. Owa, H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math., 39 (1987), 1057-1077. doi: 10.4153/CJM-1987-054-3
    [28] J. H. Park, Coefficient Estimates and Univalancy of Certain Analytic Functions, Ph. D. Thesis, Pukyong National University, Busan, 2018.
    [29] N. N. Pascu, An improvement of Becker's univalence criterion, In: Proceedings of the Commemorative Session: Simon Stoilow (Brasov, 1987), 43-48.
    [30] V. Pescar, A new generalization of Ahlfors and Becker's critetion of univalence, Bull. Malays. Math. Sci. Soc., 19 (1996), 53-54.
    [31] V. Pescar, Univalence of certain integral operators, Acta Univ. Apulensis Math. Inform., 12 (2006), 43-48.
    [32] J. A. Pfaltzgraff, Univalence of the integral of $(f'(z))^{\lambda}$, Bull. London Math. Soc., 7 (1975), 254-256.
    [33] R. Raza, S. Noreen, S. N. Malik, Geometric properties of integral operators defined by Bessel functions, J. Inequal. Spec. Funct., 7 (2016), 34-48.
    [34] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49 (1975), 109-115. doi: 10.1090/S0002-9939-1975-0367176-1
    [35] N. Seenivasagan, D. Breaz, Certain sufficient conditions for univalence, Gen. Math., 15 (2007), 7-15.
    [36] H. M. Srivastava, Operators of basic (or $q$-) calculus and fractional $q$-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. A: Sci., 44 (2020), 327-344. doi: 10.1007/s40995-019-00815-0
    [37] H. M. Srivastava, E. Deniz, H. Orhan, Some general univalence criteria for a family of integral operators, Appl. Math. Comput., 215 (2010), 3696-3701.
    [38] H. M. Srivastava, B. A. Frasin, V. Pescar, Univalence of integral operators involving Mittag-Leffler functions, Appl. Math. Inform. Sci., 11 (2017), 635-641. doi: 10.18576/amis/110301
    [39] H. M. Srivastava, R. Jan, A. Jan, W. Deebai, M. Shutaywi, Fractional-calculus analysis of the transmission dynamics of the dengue infection, Chaos, 31 (2021), Article ID 53130, 1-18.
    [40] H. M. Srivastava, A. R. S. Juma, H. M. Zayed, Univalence conditions for an integral operator defined by a generalization of the Srivastava-Attiya operator, Filomat, 32 (2018), 2101-2114. doi: 10.2298/FIL1806101S
    [41] H. M. Srivastava, S. Owa, Some characterization and distortion theorems involving fractional calculus$, $ generalized hypergeometric functions$, $ Hadamard products$, $ linear operators, and certain subclasses of analytic functions, Nagoya Math. J., 106 (1987), 1-28. doi: 10.1017/S0027763000000854
    [42] H. M. Srivastava, S. Owa, Univalent Functions$, $ Fractional Calculus$, $ and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1989.
    [43] H. M. Srivastava, J. K. Prajapat, G. I. Oros, R. Şendruţiu, Geometric properties of a certain family of integral operators, Filomat, 28 (2014), 745-754. doi: 10.2298/FIL1404745S
    [44] L. F. Stanciu, D. Breaz, H. M. Srivastava, Some criteria for univalence of a certain integral operator, Novi Sad J. Math., 43 (2013), 51-57.
    [45] N. Yağmur, Hardy space of Lommel functions, Bull. Korean Math. Soc., 52 (2015), 1035-1046. doi: 10.4134/BKMS.2015.52.3.1035
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