Subclasses of Bi-Univalent Functions Associated with q-Confluent Hypergeometric Distribution Based Upon the Horadam Polynomials

Year 2021, Volume: 5 Issue: 1, 82 - 93, 31.03.2021

Sheza El-deeb , Bassant El-matary

https://doi.org/10.31197/atnaa.768591

Cited By: 2

Abstract

In this paper, we introduce new subclasses of analytic and bi-univalent functions connected with a q-confluent hypergeometric distribution by using the Horadam polynomials which, these polynomials, the families of orthogonal polynomials and other special polynomials, as well as their extensions and generalizations, are potentially important in a variety of disciplines in many branches of science, especially in the mathematical, statistical and physical sciences. For more information associated with these polynomials . Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients |a₂| and |a₃| for functions in these subclasses and obtain Fekete-Szegő problem for these subclasses.

Keywords

Confluent hypergeometric distribution, q-derivative, Horadam polynomials, bi-univalent, coefficient bounds

References

• [1] M.H. Abu Risha, M.H. Annaby, M.E.H. Ismail and Z.S. Mansour, Linear q-difference equations, Z. Anal. Anwend., 26(2007), 481-494.
• [2] D.A. Brannan, J. Clunie and W.E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math., 22(3)(1970), 476-485.
• [3] D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions, in: S. M. Mazhar, A. Hamoui, N. S. Faour (Eds.), Mathematical Analysis and i ts Applications, Kuwait; February 18-21, 1985, in: KFAS Proceedings Series, vol. 3, Pergamon Press(Elsevier Science Limited), Oxford, 1988, pp. 53-60; see also Studia Univ. Babe³-Bolyai Math., 31(2)(1986), 70-77.
• [4] T. Bulboaca , Differential Subordinations and Superordinations. Recent Results, House of Scientific Book Publ., Cluj- Napoca, (2005).
• [5] P.L. Duren, Univalent Functions, Grundlehren der mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, (1983).
• [6] S.M. El-Deeb, Maclaurin coefficient estimates for new subclasses of bi-univalent functions connected with a q-analogue of Bessel function, Abstract Appl. Analy., (2020), Article ID 8368951, 1-7, https://doi.org/10.1155/2020/8368951.
• [7] S.M. El-Deeb, T. Bulboac and B.M. El-Matary, Maclaurin coefficient estimates of bi-univalent functions connected with the q-derivative , Mathematics, 8(2020), 1-14, https://doi.org/10.3390/math8030418.
• [8] G. Gasper and M. Rahman, Basic hypergeometric series (with a Foreword by Richard Askey). Encyclopedia of mathematics and its applications, Cambridge University Press, Cambridge, 35(1990).
• [9] A.F. Horadam, Jacobsthal representation polynomials, Fibonacci Quart. 35 (1997), 137-148.
• [10] A. F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quart. 23 (1985), 7-20.
• [11] Hörçum and E.G. Kocer, On some properties of Horadam polynomials, Internat. Math. Forum. 4 (2009), 1243-1252.
• [12] F.H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46(2)(1909), 253-281, https://doi.org/10.1017/S0080456800002751
• [13] F.H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41(1910), 193-203.
• [14] T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley Interscience Publication, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 2001.
• [15] A. Lupas, A guide of Fibonacci and Lucas polynomials, Octagon Math. Mag. 7 (1999), 2-12.
• [16] S.S. Miller and P. T. Mocanu, Differential Subordinations. Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York and Basel, (2000).
• [17] Z. Nehari, Conformal Mapping, McGraw-Hill, New York, NY, USA, (1952).
• [18] S. Porwal, Confluent hypergeometric distribution and its applications on certain classes of univalent functions of conic regions, Kyungpook Math. J., 58(2018), 495-505.
• [19] S. Porwal and S. Kumar, Confluent hypergeometric distribution and its applications on certain classes of univalent functions, Afr. Mat., 28(2017), 1-8.
• [20] E.D. Rainville, Special functions, The Macmillan Co., New York, 1960.
• [21] H.M. Srivastava, Certain q-polynomial expansions for functions of several variables. I and II, IMA J. Appl. Math. 30(1983), 205-209.
• [22] H.M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, in Univalent Functions, Fractional Calculus, and Their Applications (H. M. Srivastava and S. Owa, Editors), Halsted Press (Ellis Horwood Limited, Chichester), pp. 329-354, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1989).
• [23] 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.
• [24] H.M. Srivastava and P.W. Karlsson, Multiple Gaussian hypergeometric series, Wiley, New York, (1985).
• [25] H.M. Srivastava, A.K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23(10)(2010), 1188-1192.