Mathematica Bohemica, Vol. 148, No. 1, pp. 49-64, 2023


Faber polynomial coefficient estimates of bi-univalent functions connected with the $q$-convolution

Sheza M. El-Deeb, Serap Bulut

Received October 31, 2020.   Published online March 15, 2022.

Abstract:  We introduce a new class of bi-univalent functions defined in the open unit disc and connected with a $q$-convolution. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this class by using Faber polynomial expansions and we obtain an estimation for the Fekete-Szegö problem for this class.
Keywords:  Faber polynomial; bi-univalent function; convolution; $q$-derivative operator
Classification MSC:  05A30, 30C45, 11B65, 47B38


References:
[1] M. H. Abu Risha, M. H. Annaby, M. E. H. Ismail, Z. S. Mansour: Linear $q$-difference equations. Z. Anal. Anwend. 26 (2007), 481-494. DOI 10.4171/ZAA/1338 | MR 2341770 | Zbl 1143.39009
[2] H. Aldweby, M. Darus: On a subclass of bi-univalent functions associated with the $q$-derivative operator. J. Math. Comput. Sci., JMCS 19 (2019), 58-64. DOI 10.22436/jmcs.019.01.08
[3] M. Arif, M. Ul Haq, J.-L. Liu: A subfamily of univalent functions associated with $q$-analogue of Noor integral operator. J. Funct. Spaces 2018 (2018), Article ID 3818915, 5 pages. DOI 10.1155/2018/3818915 | MR 3762185 | Zbl 1388.30012
[4] D. A. Brannan, J. Clunie (eds.): Aspects of Contemporary Complex Analysis. Academic Press, London (1980). MR 0623462 | Zbl 0483.00007
[5] D. A. Brannan, J. Clunie, W. E. Kirwan: Coefficient estimates for a class of star-like functions. Can. J. Math. 22 (1970), 476-485. DOI 10.4153/CJM-1970-055-8 | MR 0260994 | Zbl 0197.35602
[6] D. A. Brannan, T. S. Taha: On some classes of bi-univalent functions. Stud. Univ. Babeş-Bolyai, Math. 31 (1986), 70-77. MR 0911858 | Zbl 0614.30017
[7] T. Bulboacă: Differential Subordinations and Superordinations: Recent Results. House of Scientific Book Publications, Cluj-Napoca (2005).
[8] S. Bulut: Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions. Filomat 30 (2016), 1567-1575. DOI 10.2298/FIL1606567B | MR 3530102 | Zbl 1458.30011
[9] M. Çağlar, E. Deniz: Initial coefficients for a subclass of bi-univalent functions defined by Sălăgean differential operator. Commun. Fac. Sci. Univ. Ank., Sér. A1, Math. Stat. 66 (2017), 85-91. DOI 10.1501/Commua1_0000000777 | MR 3611859 | Zbl 1391.30018
[10] M. Çağlar, H. Orhan, N. Yağmur: Coefficient bounds for new subclasses of bi-univalent functions. Filomat 27 (2013), 1165-1171. DOI 10.2298/FIL1307165C | MR 3243989 | Zbl 1324.30017
[11] P. L. Duren: Univalent Functions. Grundlehren der mathematischen Wissenschaften 259. Springer, New York (1983). MR 0708494 | Zbl 0514.30001
[12] S. M. El-Deeb: Maclaurin coefficient estimates for new subclasses of bi-univalent functions connected with a $q$-analogue of Bessel function. Abstr. Appl. Anal. 2020 (2020), Article ID 8368951, 7 pages. DOI 10.1155/2020/8368951 | MR 4104661 | Zbl 07245172
[13] S. M. El-Deeb, T. Bulboacă: Differential sandwich-type results for symmetric functions connected with a $q$-analog integral operator. Mathematics 7 (2019), Article ID 1185, 17 pages. DOI 10.3390/math7121185
[14] S. M. El-Deeb, T. Bulboacă: Fekete-Szegő inequalities for certain class of analytic functions connected with $q$-analogue of Bessel function. J. Egypt. Math. Soc. 27 (2019), Article ID 42, 11 pages. DOI 10.1186/s42787-019-0049-2 | MR 4092068 | Zbl 1435.30053
[15] S. M. El-Deeb, T. Bulboacă: Differential sandwich-type results for symmetric functions associated with Pascal distribution series. J. Contemp. Math. Anal., Armen. Acad. Sci. 56 (2021), 214-224. DOI 10.3103/S1068362321040105 | MR 4335224 | Zbl 1475.30032
[16] S. M. El-Deeb, T. Bulboacă, B. M. El-Matary: Maclaurin coefficient estimates of bi-univalent functions connected with the $q$-derivative. Mathematics 8 (2020), Article ID 418, 14 pages. DOI 10.3390/math8030418
[17] S. Elhaddad, M. Darus: Coefficient estimates for a subclass of bi-univalent functions defined by $q$-derivative operator. Mathematics 8 (2020), Article ID 306, 14 pages. DOI 10.3390/math8030306
[18] G. Faber: Über polynomische Entwickelungen. Math. Ann. 57 (1903), 389-408. (In German.) DOI 10.1007/BF01444293 | MR 1511216 | JFM 34.0430.01
[19] G. Gasper, M. Rahman: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications 35. Cambridge University Press, Cambridge (1990). DOI 10.1017/CBO9780511526251 | MR 1052153 | Zbl 0695.33001
[20] F. H. Jackson: On $q$-functions and a certain difference operator. Trans. Royal Soc. Edinburgh 46 (1909), 253-281. DOI 10.1017/S0080456800002751
[21] F. H. Jackson: On $q$-definite integrals. Quart. J. 41 (1910), 193-203. JFM 41.0317.04
[22] P. N. Kamble, M. G. Shrigan: Coefficient estimates for a subclass of bi-univalent functions defined by Sălăgean type $q$-calculus operator. Kyungpook Math. J. 58 (2018), 677-688. DOI 10.5666/KMJ.2018.58.4.677 | MR 3895894 | Zbl 1422.30018
[23] M. Lewin: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18 (1967), 63-68. DOI 10.1090/S0002-9939-1967-0206255-1 | MR 0206255 | Zbl 0158.07802
[24] S. S. Miller, P. T. Mocanu: Differential Subordinations: Theory and Applications. Pure and Applied Mathematics 225. Marcel Dekker, New York (2000). DOI 10.1201/9781482289817 | MR 1760285 | Zbl 0954.34003
[25] M. Naeem, S. Khan, F. M. Sakar: Faber polynomial coefficients estimates of bi-univalent functions. Int. J. Maps Math. 3 (2020), 57-67. MR 4179354
[26] E. Netanyahu: The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $\vert z\vert<1$. Arch. Ration. Mech. Anal. 32 (1969), 100-112. DOI 10.1007/BF00247676 | MR 0235110 | Zbl 0186.39703
[27] S. Porwal: An application of a Poisson distribution series on certain analytic functions. J. Complex Anal. 2014 (2014), Article ID 984135, 3 pages. DOI 10.1155/2014/984135 | MR 3173344 | Zbl 1310.30017
[28] J. K. Prajapat: Subordination and superordination preserving properties for generalized multiplier transformation operator. Math. Comput. Modelling 55 (2012), 1456-1465. DOI 10.1016/j.mcm.2011.10.024 | MR 2887529 | Zbl 1255.30024
[29] F. M. Sakar, M. Naeem, S. Khan, S. Hussain: Hankel determinant for class of analytic functions involving $q$-derivative operator. J. Adv. Math. Stud. 14 (2021), 265-278. Zbl 07389042
[30] H. M. Srivastava: Certain $q$-polynomial expansions for functions of several variables. IMA J. Appl. Math. 30 (1983), 315-323. DOI 10.1093/imamat/30.3.315 | MR 0719983 | Zbl 0504.33003
[31] H. M. Srivastava: Certain $q$-polynomial expansions for functions of several variables. II. IMA J. Appl. Math. 33 (1984), 205-209. DOI 10.1093/imamat/33.2.205 | MR 0767521 | Zbl 0544.33007
[32] H. M. Srivastava: Univalent functions, fractional calculus, and associated generalized hypergeometric functions. Univalent Functions, Fractional Calculus, and Their Applications John Willey & Sons, New York (1989), 329-354. MR 1199160 | Zbl 0693.30013
[33] 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 | MR 4064730
[34] H. M. Srivastava, S. Bulut, M. Çağlar, N. Yağmur: Coefficient estimates for a general subclass of analytic and bi-univalent functions. Filomat 27 (2013), 831-842. DOI 10.2298/FIL1305831S | MR 3186102 | Zbl 1432.30014
[35] H. M. Srivastava, S. S. Eker, R. M. Ali: Coefficient bounds for a certain class of analytic and bi-univalent functions. Filomat 29 (2015), 1839-1845. DOI 10.2298/FIL1508839S | MR 3403901 | Zbl 1458.30035
[36] H. M. Srivastava, S. S. Eker, S. G. Hamidi, J. M. Jahangiri: Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator. Bull. Iran. Math. Soc. 44 (2018), 149-157. DOI 10.1007/s41980-018-0011-3 | MR 3879475 | Zbl 1409.30021
[37] H. M. Srivastava, S. M. El-Deeb: A certain class of analytic functions of complex order with a $q$-analogue of integral operators. Miskolc Math. Notes 21 (2020), 417-433. DOI 10.18514/MMN.2020.3102 | MR 4133288 | Zbl 07254908
[38] H. M. Srivastava, S. M. El-Deeb: 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 (2020), 7087-7106. DOI 10.3934/math.2020454 | MR 4150209
[39] H. M. Srivastava, P. W. Karlsson: Multiple Gaussian Hypergeometric Series. Ellis Horwood Series in Mathematics and Its Applications. John Wiley & Sons, New York (1985). MR 0834385 | Zbl 0552.33001
[40] H. M. Srivastava, S. Khan, Q. Z. Ahmad, N. Khan, S. Hussain: 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. Babeş-Bolyai, Math. 63 (2018), 419-436. DOI 10.24193/subbmath.2018.4.01 | MR 3886058 | Zbl 1438.05021
[41] H. M. Srivastava, A. K. Mishra, P. Gochhayat: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23 (2010), 1188-1192. DOI 10.1016/j.aml.2010.05.009 | MR 2665593 | Zbl 1201.30020
[42] H. M. Srivastava, A. Motamednezhad, E. A. Adegani: Faber polynomial coefficient estimates for bi-univalent functions defined by using differential subordination and a certain fractional derivative operator. Mathematics 8 (2020), Article ID 172, 12 pages. DOI 10.3390/math8020172
[43] H. M. Srivastava, G. Murugusundaramoorthy, S. M. El-Deeb: Faber polynomial coefficient estimates of bi-close-convex functions connected with the Borel distribution of the Mittag-Leffler type. J. Nonlinear Var. Anal. 5 (2021), 103-118. DOI 10.23952/jnva.5.2021.1.07 | Zbl 07312309
[44] H. M. Srivastava, F. M. Sakar, H. O. Güney: Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination. Filomat 32 (2018), 1313-1322. DOI 10.2298/FIL1804313S | MR 3848107 | Zbl 07462770

Affiliations:   Sheza Mohammed El-Deeb, Department of Mathematics, College of Science and Arts in Al-Badaya, Qassim University, Buraidah 51452, Saudi Arabia and Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt, e-mail: shezaeldeeb@yahoo.com; Serap Bulut, Kocaeli University, Faculty of Aviation and Space Sciences, Arslanbey Campus, 41285 Kartepe-Kocaeli, Turkey, e-mail: serap.bulut@kocaeli.edu.tr


 
PDF available at: