| Title:
|
Faber polynomial coefficient estimates of bi-univalent functions connected with the $q$-convolution (English) |
| Author:
|
El-Deeb, Sheza M. |
| Author:
|
Bulut, Serap |
| Language:
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English |
| Journal:
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Mathematica Bohemica |
| ISSN:
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0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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148 |
| Issue:
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1 |
| Year:
|
2023 |
| Pages:
|
49-64 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Faber polynomial |
| Keyword:
|
bi-univalent function |
| Keyword:
|
convolution |
| Keyword:
|
$q$-derivative operator |
| MSC:
|
05A30 |
| MSC:
|
11B65 |
| MSC:
|
30C45 |
| MSC:
|
47B38 |
| idZBL:
|
Zbl 07655812 |
| idMR:
|
MR4536309 |
| DOI:
|
10.21136/MB.2022.0173-20 |
| . |
| Date available:
|
2023-02-03T11:21:27Z |
| Last updated:
|
2023-09-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151526 |
| . |
| Reference:
|
[1] Risha, M. H. Abu, Annaby, M. H., Ismail, M. E. H., Mansour, Z. S.: Linear $q$-difference equations.Z. Anal. Anwend. 26 (2007), 481-494. Zbl 1143.39009, MR 2341770, 10.4171/ZAA/1338 |
| Reference:
|
[2] Aldweby, H., Darus, M.: On a subclass of bi-univalent functions associated with the $q$-derivative operator.J. Math. Comput. Sci., JMCS 19 (2019), 58-64. 10.22436/jmcs.019.01.08 |
| Reference:
|
[3] Arif, M., Haq, M. Ul, Liu, J.-L.: A subfamily of univalent functions associated with $q$-analogue of Noor integral operator.J. Funct. Spaces 2018 (2018), Article ID 3818915, 5 pages. Zbl 1388.30012, MR 3762185, 10.1155/2018/3818915 |
| Reference:
|
[4] Brannan, D. A., (eds.), J. Clunie: Aspects of Contemporary Complex Analysis.Academic Press, London (1980). Zbl 0483.00007, MR 0623462 |
| Reference:
|
[5] Brannan, D. A., Clunie, J., Kirwan, W. E.: Coefficient estimates for a class of star-like functions.Can. J. Math. 22 (1970), 476-485. Zbl 0197.35602, MR 0260994, 10.4153/CJM-1970-055-8 |
| Reference:
|
[6] Brannan, D. A., Taha, T. S.: On some classes of bi-univalent functions.Stud. Univ. Babeş-Bolyai, Math. 31 (1986), 70-77. Zbl 0614.30017, MR 0911858 |
| Reference:
|
[7] Bulboacă, T.: Differential Subordinations and Superordinations: Recent Results.House of Scientific Book Publications, Cluj-Napoca (2005). |
| Reference:
|
[8] Bulut, S.: Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions.Filomat 30 (2016), 1567-1575. Zbl 1458.30011, MR 3530102, 10.2298/FIL1606567B |
| Reference:
|
[9] Çağlar, M., Deniz, E.: 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. Zbl 1391.30018, MR 3611859, 10.1501/Commua1_0000000777 |
| Reference:
|
[10] Çağlar, M., Orhan, H., Yağmur, N.: Coefficient bounds for new subclasses of bi-univalent functions.Filomat 27 (2013), 1165-1171. Zbl 1324.30017, MR 3243989, 10.2298/FIL1307165C |
| Reference:
|
[11] Duren, P. L.: Univalent Functions.Grundlehren der mathematischen Wissenschaften 259. Springer, New York (1983). Zbl 0514.30001, MR 0708494 |
| Reference:
|
[12] El-Deeb, S. M.: 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. Zbl 07245172, MR 4104661, 10.1155/2020/8368951 |
| Reference:
|
[13] El-Deeb, S. M., Bulboacă, T.: Differential sandwich-type results for symmetric functions connected with a $q$-analog integral operator.Mathematics 7 (2019), Article ID 1185, 17 pages. 10.3390/math7121185 |
| Reference:
|
[14] El-Deeb, S. M., Bulboacă, T.: 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. Zbl 1435.30053, MR 4092068, 10.1186/s42787-019-0049-2 |
| Reference:
|
[15] El-Deeb, S. M., Bulboacă, T.: Differential sandwich-type results for symmetric functions associated with Pascal distribution series.J. Contemp. Math. Anal., Armen. Acad. Sci. 56 (2021), 214-224. Zbl 1475.30032, MR 4335224, 10.3103/S1068362321040105 |
| Reference:
|
[16] 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), Article ID 418, 14 pages. 10.3390/math8030418 |
| Reference:
|
[17] Elhaddad, S., Darus, M.: Coefficient estimates for a subclass of bi-univalent functions defined by $q$-derivative operator.Mathematics 8 (2020), Article ID 306, 14 pages. 10.3390/math8030306 |
| Reference:
|
[18] Faber, G.: Über polynomische Entwickelungen.Math. Ann. 57 (1903), 389-408 \hbox{German} \99999JFM99999 34.0430.01. MR 1511216, 10.1007/BF01444293 |
| Reference:
|
[19] Gasper, G., Rahman, M.: Basic Hypergeometric Series.Encyclopedia of Mathematics and Its Applications 35. Cambridge University Press, Cambridge (1990). Zbl 0695.33001, MR 1052153, 10.1017/CBO9780511526251 |
| Reference:
|
[20] Jackson, F. H.: On $q$-functions and a certain difference operator.Trans. Royal Soc. Edinburgh 46 (1909), 253-281. 10.1017/S0080456800002751 |
| Reference:
|
[21] Jackson, F. H.: On $q$-definite integrals.Quart. J. 41 (1910), 193-203 \99999JFM99999 41.0317.04. |
| Reference:
|
[22] Kamble, P. N., Shrigan, M. G.: Coefficient estimates for a subclass of bi-univalent functions defined by Sălăgean type $q$-calculus operator.Kyungpook Math. J. 58 (2018), 677-688. Zbl 1422.30018, MR 3895894, 10.5666/KMJ.2018.58.4.677 |
| Reference:
|
[23] Lewin, M.: On a coefficient problem for bi-univalent functions.Proc. Am. Math. Soc. 18 (1967), 63-68. Zbl 0158.07802, MR 0206255, 10.1090/S0002-9939-1967-0206255-1 |
| Reference:
|
[24] Miller, S. S., Mocanu, P. T.: Differential Subordinations: Theory and Applications.Pure and Applied Mathematics 225. Marcel Dekker, New York (2000). Zbl 0954.34003, MR 1760285, 10.1201/9781482289817 |
| Reference:
|
[25] Naeem, M., Khan, S., Sakar, F. M.: Faber polynomial coefficients estimates of bi-univalent functions.Int. J. Maps Math. 3 (2020), 57-67. MR 4179354 |
| Reference:
|
[26] Netanyahu, E.: 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. Zbl 0186.39703, MR 0235110, 10.1007/BF00247676 |
| Reference:
|
[27] Porwal, S.: An application of a Poisson distribution series on certain analytic functions.J. Complex Anal. 2014 (2014), Article ID 984135, 3 pages. Zbl 1310.30017, MR 3173344, 10.1155/2014/984135 |
| Reference:
|
[28] Prajapat, J. K.: Subordination and superordination preserving properties for generalized multiplier transformation operator.Math. Comput. Modelling 55 (2012), 1456-1465. Zbl 1255.30024, MR 2887529, 10.1016/j.mcm.2011.10.024 |
| Reference:
|
[29] Sakar, F. M., Naeem, M., Khan, S., Hussain, S.: Hankel determinant for class of analytic functions involving $q$-derivative operator.J. Adv. Math. Stud. 14 (2021), 265-278. Zbl 07389042, MR 4398168 |
| Reference:
|
[30] Srivastava, H. M.: Certain $q$-polynomial expansions for functions of several variables.IMA J. Appl. Math. 30 (1983), 315-323. Zbl 0504.33003, MR 0719983, 10.1093/imamat/30.3.315 |
| Reference:
|
[31] Srivastava, H. M.: Certain $q$-polynomial expansions for functions of several variables. II.IMA J. Appl. Math. 33 (1984), 205-209. Zbl 0544.33007, MR 0767521, 10.1093/imamat/33.2.205 |
| Reference:
|
[32] Srivastava, H. M.: Univalent functions, fractional calculus, and associated generalized hypergeometric functions.Univalent Functions, Fractional Calculus, and Their Applications John Willey & Sons, New York (1989), 329-354. Zbl 0693.30013, MR 1199160 |
| Reference:
|
[33] 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. A, Sci. 44 (2020), 327-344. MR 4064730, 10.1007/s40995-019-00815-0 |
| Reference:
|
[34] Srivastava, H. M., Bulut, S., Çağlar, M., Yağmur, N.: Coefficient estimates for a general subclass of analytic and bi-univalent functions.Filomat 27 (2013), 831-842. Zbl 1432.30014, MR 3186102, 10.2298/FIL1305831S |
| Reference:
|
[35] Srivastava, H. M., Eker, S. S., Ali, R. M.: Coefficient bounds for a certain class of analytic and bi-univalent functions.Filomat 29 (2015), 1839-1845. Zbl 1458.30035, MR 3403901, 10.2298/FIL1508839S |
| Reference:
|
[36] Srivastava, H. M., Eker, S. S., Hamidi, S. G., Jahangiri, J. M.: Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator.Bull. Iran. Math. Soc. 44 (2018), 149-157. Zbl 1409.30021, MR 3879475, 10.1007/s41980-018-0011-3 |
| Reference:
|
[37] 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), 417-433. Zbl 07254908, MR 4133288, 10.18514/MMN.2020.3102 |
| Reference:
|
[38] 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 (2020), 7087-7106. MR 4150209, 10.3934/math.2020454 |
| Reference:
|
[39] Srivastava, H. M., Karlsson, P. W.: Multiple Gaussian Hypergeometric Series.Ellis Horwood Series in Mathematics and Its Applications. John Wiley & Sons, New York (1985). Zbl 0552.33001, MR 0834385 |
| Reference:
|
[40] 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. Babeş-Bolyai, Math. 63 (2018), 419-436. Zbl 1438.05021, MR 3886058, 10.24193/subbmath.2018.4.01 |
| Reference:
|
[41] Srivastava, H. M., Mishra, A. K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions.Appl. Math. Lett. 23 (2010), 1188-1192. Zbl 1201.30020, MR 2665593, 10.1016/j.aml.2010.05.009 |
| Reference:
|
[42] Srivastava, H. M., Motamednezhad, A., Adegani, E. A.: 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. 10.3390/math8020172 |
| Reference:
|
[43] Srivastava, H. M., Murugusundaramoorthy, G., El-Deeb, S. M.: 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. Zbl 07312309, 10.23952/jnva.5.2021.1.07 |
| Reference:
|
[44] Srivastava, H. M., Sakar, F. M., Güney, H. O.: Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination.Filomat 32 (2018), 1313-1322. Zbl 07462770, MR 3848107, 10.2298/FIL1804313S |
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