Coefficient estimates for a class of bi-univalent functions involving Mittag-Leffler type Borel distribution

Volume 16, Issue 3, pp 180--197 http://dx.doi.org/10.22436/jnsa.016.03.04

Publication Date: October 08, 2023 Submission Date: July 31, 2023 Revision Date: August 17, 2023 Accteptance Date: August 20, 2023

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 178 Downloads
• 320 Views

Authors

B. Khan - School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, Peoples Republic of China. M. Ghaffar Khan - Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan. T. G. Shaba - Department of Mathematics, Landmark University, Omu-Aran 251103, Nigeria.

Abstract

In recent years, many new subclasses of analytic and bi-univalent functions have been studied and examined from different viewpoints and prospectives. In this article, we introduce new subclass of analytic and bi-univalent functions based on Mittag-Leffler type Borel distribution associated with the Gegenbauer polynomials. Furthermore we obtain estimates for \(\left\vert a_{2}\right\vert ,\) \(\left\vert a_{3}\right\vert \), and \(\left\vert a_{4}\right\vert \) coefficients and Fekete-Szego inequality for this functions class. Providing specific values to parameters involved in our main results, we get some new results.

Share and Cite

Keywords

• Analytic function
• bi-univalent function
• Gegenbaure polynomials
• coefficient estimates
• subordination
• Fekete-Szego functional problems

MSC

•  30C45
•  30C50
•  30C80

References

• [1] B. Ahmad, M. G. Khan, B. A. Frasin, M. K. Aouf, T. Abdeljawad, W. K. Mashwani, M. Arif, On q-analogue of meromorphic multivalent functions in lemniscate of Bernoulli domain, AIMS Math., 6 (2021), 3037–3052
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] I. Aldawish, T. Al-Hawary, B. A. Frasin, Subclasses of Bi-Univalent Functions Defined by Frasin Differential Operator, Mathematics, 8 (2020), 11 pages
  • View Article


• [3] W. A. Al-Salam, L. Carlitz, The Gegenbauer addition theorem, J. Math. Phys., 42 (1963), 147–156
  • View Article
  • Google Scholar
  • MATH


• [4] S¸. Altınkaya, S. Yalc¸ın, On the (p, q)-Lucas polynomial coefficient bounds of the bi-univalent function class , Bol. Soc. Mat. Mex. (3), 25 (2019), 567–575
  • View Article
  • MathSciNet
  • Google Scholar


• [5] S¸. Altınkaya, S. Yalc¸ın, The (p, q)-Chebyshev polynomial bounds of a general bi-univalent function class, Bol. Soc. Mat. Mex. (3), 26 (2020), 341–348
  • View Article
  • MathSciNet
  • Google Scholar


• [6] A. Amourah, A. Alamoush, M. Al-Kaseasbeh, Gegenbauer polynomials and bi-univalent functions, Palest. J. Math., 10 (2021), 625–632
  • View Article
  • Google Scholar
  • MATH


• [7] A. Attiya, Some applications of Mittag-Leffler function in the unit disk, Filomat, 30 (2016), 2075–2081
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] D. Bansal, J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ., 61 (2016), 338–350
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys., 18 (2011), 133–143
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] D. A. Brannan, J. G. Clunie, Aspects of contemporary complex analysis, Academic Press, London-New York (1980)
  • View Article
  • Google Scholar


• [11] D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes¸-Bolyai Math., 31 (1986), 70–77
  • View Article
  • Google Scholar


• [12] S. Bulut, Coefficient estimates for a class of analytic and bi-univalent functions, Novi Sad J. Math., 43 (2013), 59–65
  • View Article


• [13] V. S. Buyarov, P. L´opez-Art´es, A. Mart´ınez-Finkelshtein, W. Van Assche, Information entropy of Gegenbauer polynomials, J. Phys. A, 33 (2000), 6549–6560
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] S. M. El-Deeb, G. Murugusundaramoorthy, A. Alburaikan, Bi-Bazilevic functions based on the Mittag-Leffler-type Borel distribution associated with Legendre polynomials, J. Math. Comput. Sci., 24 (2022), 235–245
  • View Article
  • Google Scholar


• [15] P. Filipponi, A. F. Horadam, Second derivative sequences of Fibonacci and Lucas polynomials, Fibonacci Quart., 31 (1993), 194–204
  • View Article
  • Google Scholar
  • MATH


• [16] B. A. Frasin, Coefficient bounds for certain classes of bi-univalent functions, Hacet. J. Math. Stat., 43 (2014), 383–389
  • View Article
  • Google Scholar
  • MATH


• [17] B. A. Frasin, An application of an operator associated with generalized Mittag-Leffler function, Konuralp J. Math., 7 (2019), 199–202
  • View Article
  • Google Scholar
  • MATH


• [18] B. A. Frasin, T. Al-Hawary, F. Yousef, Some properties of a linear operator involving generalized Mittag-Leffler function, Stud. Univ. Babes¸-Bolyai Math., 65 (2020), 67–75
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24 (2011), 1569–1573
  • View Article
  • MathSciNet
  • Google Scholar


• [20] M. Fekete, G. Szego, Eine Bemerkinkung A˜ berunggerade schlichte Funktionen, J. Lond. Math. Soc., s1-8 (1933), 85–89
  • View Article


• [21] M. Garg, P. Manohar, S. L. Kalla, A Mittag-Leffler-typefunction of two variables, Integral Transforms Spec. Funct., 24 (2013), 934–944
  • View Article
  • Google Scholar


• [22] Q. Hu, H. M. Srivastava, B. Ahmad, N. Khan, M. G. Khan, W. K. Mashwani, B. Khan, A subclass of multivalent Janowski type q-starlike functions and its consequences, Symmetry, 13 (2021), 1–14
  • View Article
  • Google Scholar


• [23] M. G. Khan, B. Ahmad, N. Khan, W. K. Mashwani, S. Arjika, B. Khan, R. Chinram, Applications of Mittag-Leffler type Poisson distribution to a subclass of analytic functions involving conic-type regions, J. Funct. Spaces, 2021 (2021), 9 pages
  • View Article
  • Google Scholar


• [24] B. Khan, I. Aldawish, S. Araci, M. G. Khan, Third Hankel determinant for the logarithmic coefficients of starlike functions associated with sine function, Fractal Fract., 6 (2022), 1–11
  • View Article
  • Google Scholar


• [25] S. Khan, A. A. Al-Gonah, G. Yasmin, Generalized and mixed type Gegenbauer polynomials, J. Math. Anal. Appl., 390 (2012), 197–207
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] B. Khan, Z.-G. Liu, T. G. Shaba, S. Araci, N. Khan, M. G. Khan, Applications of q-derivative operator to the subclass of bi-univalent functions involving q-Chebyshev polynomials, J. Math., 2022 (2022), 7 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [27] B. Khan, Z.-G. Liu, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, A study of some families of multivalent q-starlike functions involving higher-order q-Derivatives, Mathematics, 8 (2020), 1–12
  • View Article
  • Google Scholar


• [28] B. Khan, H. M. Srivastava, N. Khan, M. Darus, Q. Z. Ahmad, M. Tahir, Applications of certain conic domains to a subclass of q-starlike functions associated with the Janowski functions, Symmetry, 13 (2021), 1–18
  • View Article
  • Google Scholar


• [29] K. R. Karthikeyan, G. Murugusundaramoorthy, S. D. Purohit, D. L. Suthar, Certain class of analytic functions with respect to symmetric points defined by Q-calculus, J. Math., 2021 (2021), 9 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] G. Y. Lee, M. Asci, Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials, J. Appl. Math., 2012 (2012), 18 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [31] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63–68
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] A. Lupas, A guide of Fibonacci and Lucas polynomials, Octogon Math. Mag., 7 (1999), 3–12
  • View Article
  • Google Scholar


• [33] G. M. Mittag-Leffler, Sur la nouvelle fonction E(x), C. R. Acad. Sci. Paris, 137 (1903), 554–558
  • View Article
  • Google Scholar
  • MATH


• [34] G. Murugusundaramoorthy, S. M. El-Deeb, Second Hankel determinant for a class of analytic functions of the Mittag- Leffler-type Borel distribution related with Legendre polynomials, TWMS J. Appl. Eng. Math., 12 (2022), 1247–1258
  • View Article
  • Google Scholar


• [35] E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in z < 1, Arch. Rational Mech. Anal., 32 (1969), 100–112
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [36] S. D. Purohit, M. M. Gour, S. Joshi, D. L. Suthar, Certain classes of analytic functions bound with Kober operators in q-calculus, J. Math., 2021 (2021), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [37] C. Ramachandran, S. Annamalai, B. A. Frasin, The q-difference operator associated with the multivalent function bounded by conical sections, Bol. Soc. Parana. Mat. (3), 39 (2021), 133–146
  • View Article
  • Google Scholar
  • MATH


• [38] J. S´anchez-Ruiz, Information entropy of Gegenbauer polynomials and Gaussian quadrature, J. Phys. A, 36 (2003), 4857–4865
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [39] K. A. Selvakumaran, M. Gomathi, S. D. Purohit, D. Kumar, Fekete-SzegA˜ ¶ Inequalities for New Classes of Analytic Functions Associated with Fractional q-Differintegral Operator, Sci. Technol. Asia, 26 (2021), 160-168
  • View Article


• [40] M. Shah, Applications of Gegenbauer (ultraspherical) polynomials in cooling of a heated cylinder, An. Univ. Timis¸oara Ser. S¸ti. Mat., 8 (1970), 207–212
  • View Article
  • Google Scholar


• [41] L. Shi, B. Ahmad, N. Khan, M. G. Khan, S. Araci, W. K. Mashwani, B. Khan, Coefficient estimates for a subclass of meromorphic multivalent q-close-to-convex functions, Symmetry, 13 (2021), 1–12
  • View Article
  • Google Scholar


• [42] L. Shi, M. G. Khan, B. Ahmad, Some geometric properties of a family of analytic functions involving a generalized q-operator, Symmetry, 12 (2020), 1–11
  • View Article
  • Google Scholar


• [43] I. N. Sneddon, The evaluation of an integral involving the product of two Gegenbauer polynomials, SIAM Rev., 9 (1967), 569–572
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [44] 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. Sci., 44 (2020), 327–344
  • View Article
  • Google Scholar


• [45] H. M. Srivastava, Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformatioons, J. Nonlinear Convex Anal., 22 (2021), 1501–1520
  • View Article
  • Google Scholar


• [46] H. M. Srivastava, B. Khan, N. Khan, M. Tahir, S. Ahmad, N. Khan, Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function, Bull. Sci. Math., 167 (2021), 16 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [47] T. S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, (1981),
  • View Article
  • Google Scholar


• [48] P. Vellucci, A. M. Bersani, The class of Lucas-Lehmer polynomials, Rend. Mat. Appl. (7), 37 (2016), 43–62
  • View Article
  • Google Scholar
  • MATH


• [49] A. K. Wanas, J. A. Khuttar, Applications of Borel distribution series on analytic functions, Earthline J. Math. Sci., 4 (2020), 71–82
  • View Article
  • Google Scholar


• [50] T. Wang, W. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55 (2012), 95–103
  • View Article
  • Google Scholar
  • MATH


• [51] A. Wiman, U¨ ber die Nullstellen der Funktionen E(x), Acta Math., 29 (1905), 217–234
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [52] C. Zhang, B. Khan, T. G. Shaba, J.-S. Ro, S. Araci, M. G. Khan, Applications of q-Hermite polynomials to subclasses of analytic and bi-univalent Functions, Fractal Fract., 6 (2022), 1–15
  • View Article
  • Google Scholar


• [53] H. Zhou, K. A. Selvakumaran, S. Sivasubramanian, S. D. Purohit, H. Tang, Subordination problems for a new class of Bazileviˇc functions associated with k-symmetric points and fractional q-calculus operators, AIMS Math., 6 (2021), 8642–8653
  • View Article
  • Google Scholar
  • MATH