Starlikeness of an integral operator associated with Mittag-Leffler functions

Document Type : Research Paper

Authors

1 Department of Mathematics UIET, CSJM University, Kanpur-208024, (U.P.), India

2 Department of Mathematics, Ram Sahai Government Degree College, Bairi-Shivrajpur, Kanpur-209205, (Uttar Pradesh), India

3 Department of Mathematics, Government Engineering College-Dahod Gujarat-389151, India

Abstract

In the present paper, we introduce a new integral operator involving with Mittag-Leffler function and the Salagean operator. Further, we obtain some sufficient conditions for this integral operator belonging to certain classes of starlike functions.

Keywords

[1] M. Arif and M. Raza, Some properties of an integral operator defined by Bessel functions, Acta Univ. Apulensis 26 (2011), 69–74.
[2] A.A. Attiya, Some applications of Mittag-Leffler function in the unit disk, Filomat 30 (2016), no. 7, 2075–2081.
[3] D. Bansal and J.K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ. 61 (2016), no. 3, 338–350.
[4] D. Breaz, Certain integral operators on the classes M(βi) and N(βi), J. Inequal. Appl. 2008 (2008), Art. ID 719354, 1–4.
[5] E. Deniz, H. Orhan and 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.
[6] B.A. Frasin, Sufficient condition for integral operator defined by Bessel functions, J. Math. Inequal. 4 (2010), no. 3, 301–306.
[7] H.O. G¨uney, G.I. Oros and S. Owa, ¨ An application of Salagean operator concerning starlike functions, Axioms 11 (2022), no. 2, 50. https://doi.org/10.3390/axioms11020050
[8] A.R.S. Juma and L.I. Cotirla, On harmonic univalent function defined by generalized Salagean derivatives, Acta Univ. Apulensis 23 (2010), 179–188.
[9] E. Kadioˇglu, On subclass of univalent functions with negative coefficients, Appl. Math. Comput. 146 (2003), 351–358.
[10] A.A. Lupa¸s, On special fuzzy differential subordinations obtained for Riemann–Liouville fractional integral of Ruscheweyh and S˘al˘agean operators, Axioms 11 (2022), no. 9, 428.
[11] N. Magesh, S. Porwal and S.P. Singh, Some geometric properties of an integral operator involving Bessel functions, Novi Sad J. Math. 47 (2017), no. 2, 149–156.
[12] S. Mahmood, H.M. Srivastava, S.N. Malik, M. Raza, N. Shahzadi and S. Zainab, A certain family of integral operators associated with the Struve functions, Symmetry 11 (2019), Art. ID 463, 1–16.
[13] G.M. Mittag-Leffler, Sur la nouvelle function E(x), C. R. Acad. Sci. Paris 137 (1903), 554–558.
[14] N.N. Pasai and V. Pescar, On the integral operators of Kim-merkes and Pfaltzgraff, Mathematica 32 (1990), no. 2, 185–192.
[15] G.H. Park, H.M. Srivastava and N.E. Cho, Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind, AIMS Math. 6 (2021), no. 10, 11380—11402.
[16] Saurabh Porwal, Mapping properties of an integral operator, Acta Univ. Apulensis 27 (2011), 151–155.
[17] Saurabh Porwal, Geometric properties of an integral operator associated with Bessel functions, Electronic J. Math. Anal. Appl. 8 (2020), no. 2, 75–80.
[18] S. Porwal and D. Breaz, Mapping properties of an integral operator involving Bessel functions, Analytic Number Theory, Approximation Theory and Spect. Funct., 821-826, Springer, New York, 2014.
[19] S. Porwal and M. Kumar, Mapping properties of an integral operator involving Bessel functions on some subclasses of univalent functions, Afr. Mat. 28 (2017), no. 1-2, 165–170
[20] S. Porwal, A. Gupta and G. Murugasundaramoorthy, New sufficient conditions for starlikeness of certain integral operators involving Bessel functions, Acta Univ. Math. Belii Ser. Math. 27 (2017), 10–18.
[21] M. Raza, S. Noreen and S.N. Malik, Geometric properties of integral operators defined by Bessel functions, J. Ineq. Spec. Funct., 7(2016), 34-48.
[22] M. S. Robertson, On the theory of univalent functions, Ann. Math. 37 (1936), no. 2, 374–408.
[23] G.S. Salagean, Subclasses of univalent functions, Complex Anal. Fifth Roman. Finish Seminar, Bucharest, 1983, pp. 362–372.
[24] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109–116.
[25] H. Shiraishi and S. Owa, Starlikeness and convexity for analytic functions concerned with Jack’s Lemma, Int. J. Open Problem Comput. Math. 2 (2009), no. 1, 37–47.
[26] H.M. Srivastava, B.A. Frasin and V. Pescar, Univalance of integral operators involving Mittag-Leffler functions, Appl. Math. Inf. Sci. 11 (2017), no. 3, 635–641.
[27] A. Wiman, Uber den fundamental satz in der Theorie der Funcktionen ˘ E(x), Acta Math. 29 (1905), 191–201.
[28] A. Wiman, Uber die Nullstellum der Funcktionen ˘ E(x), Acta Math. 29 (1905), 271–234.
Volume 14, Issue 5
May 2023
Pages 1-7
  • Receive Date: 21 July 2022
  • Revise Date: 09 February 2023
  • Accept Date: 24 February 2023