New subclasses of bi-univalent functions connected with a q-analogue of convolution based upon the Legendre polynomials
DOI:
https://doi.org/10.24193/subbmath.2023.3.06Keywords:
Legendre polynomials, convolution, q-analogue of Pascal distribution, q-analogue of poission operator, bi-univalent, coefficients bounds.Abstract
In this paper, we introduce new subclasses of analytic and bi-univalent functions connected with a q-analogue of convolution by using the Legendre poly- nomials. 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.
Mathematics Subject Classification (2010): 30C50, 30C45, 11B65, 47B38.
References
Abu Risha, M.H., Annaby, M.H., Ismail, M.E.H., Mansour, Z.S., Linear q-difference equations, Z. Anal. Anwend., 26(2007), 481-494.
Arif, M., Ul Haq, M., Liu, J.L., A subfamily of univalent functions associated with q- analogue of Noor integral operator, J. Function Spaces, (2018), Art. ID 3818915, 1-5, https://doi.org/10.1155/2018/3818915.
Brannan, D.A., Clunie, J., Kirwan, W.E., Coefficient estimates for a class of starlike functions, Canad. J. Math., 22(3)(1970), 476-485.
Brannan, D.A., Taha, T.S., On some classes of bi-univalent functions, in: S. M. Mazhar, A. Hamoui, N. S. Faour (Eds.), Mathematical Analysis and its 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. Babeș-Bolyai Math., 31(2)(1986), 70-77.
Bulboacă, T., Differential Subordinations and Superordinations, Recent Results, House of Scientific Book Publ., Cluj-Napoca, 2005.
Duren, P.L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
El-Deeb, S.M., 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.
El-Deeb, S.M., Bulboacă, T., Fekete-Szegő inequalities for certain class of analytic functions connected with q-anlogue of Bessel function, J. Egyptian. Math. Soc., (2019), 1-11, https://doi.org/10.1186/s42787-019-0049-2.
El-Deeb, S.M., Bulboacă, T., Differential sandwich-type results for symmetric functions connected with a q-analog integral operator, Mathematics, 7(2019), no. 12, 1-17, https://doi.org/10.3390/math7121185.
El-Deeb, S.M., Bulboacă, T., Differential sandwich-type results for symmetric functions associated with Pascal distribution series, J. Contemporary Math. Anal. (in press).
El-Deeb, S.M., Bulboacă, T., Dziok, J., Pascal distribution series connected with certain subclasses of univalent functions, Kyungpook Math. J., 59(2019), 301-314.
El-Deeb, S.M., Bulboacă, T., El-Matary, B.M., Maclaurin coefficient estimates of bi-univalent functions connected with the q-derivative, Mathematics, 8(2020), 1-14, https://doi.org/10.3390/math8030418.
Gasper, G., Rahman, M., Basic Hypergeometric Series (with a Foreword by Richard Askey), Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 35, 1990.
Jackson, F.H., On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46(1909), no. 2, 253-281, https://doi.org/10.1017/S0080456800002751
Jackson, F.H., On q-definite integrals, Quart. J. Pure Appl. Math., 41(1910), 193-203.
Lebedev, N., Special Functions and Their Applications, Dover, New York, 1972.
Miller, S.S., Mocanu, P.T., 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.
Nehari, Z., Conformal Mapping, McGraw-Hill, New York, NY, 1952.
Porwal, S., An application of a Poisson distribution series on certain analytic functions, J. Complex Anal., (2014), Art. ID 984135, 1-3, https://dx.doi.org/10.1155/2014/984135.
Prajapat, J.K., Subordination and superordination preserving properties for generalized multiplier transformation operator, Math. Comput. Modelling, 55(2012), 1456-1465.
Srivastava, H.M., Certain q-polynomial expansions for functions of several variables, I and II, IMA J. Appl. Math., 30(1983), 205-209.
Srivastava, H.M., Univalent functions, fractional calculus, and associated generalized hypergeometric functions, in Univalent Functions, Fractional Calculus, and Their Ap- plications (H.M. Srivastava and S. Owa, Editors), Halsted Press (Ellis Horwood Limited, Chichester), 329-354, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989.
Srivastava, H.M., 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.
Srivastava, H.M., El-Deeb, S.M., A certain class of analytic functions of complex order with a q-analogue of integral operators, Miskolc Math. Notes, 21(2020), no. 1, 417-433.
Srivastava, H.M., El-Deeb, S.M., The Faber polynomial expansion method and the Taylor-Maclaurin coefficient estimates of bi-close-to-convex functions connected with the q-convolution, AIMS Math., 5(6)(2020), 7087-7106.
Srivastava, H.M., Karlsson, P.W., Multiple Gaussian Hypergeometric Series, Wiley, New York, 1985.
Srivastava, H.M., Khan, S., Ahmad, Q.Z., Khan, N., Hussain, S., The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain q-integral operator, Stud. Univ. Babeș-Bolyai Math., 63(2018), 419-436.
Srivastava, H.M., Mishra, A.K., Gochhayat, P., Certain subclasses of analytic and bi- univalent functions, Appl. Math. Lett., 23(10)(2010), 1188-1192.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
