| Title:
|
Coefficient bounds for the functions and inverse functions in the newly constructed subclass of bi-univalent functions (English) |
| Author:
|
Aşır, Gülizar |
| Author:
|
Aktaş, İbrahim |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0011-4642 |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
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151 |
| Issue:
|
4 |
| Year:
|
2026 |
| Pages:
|
603-617 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We introduce a novel subclass of bi-univalent functions with respect to symmetric points using generalized bivariate Fibonacci polynomials. Then we determine certain bounds for the initial coefficients of Maclaurin expansion of the functions belonging to this class of functions. In addition, we study the well-known Fekete-Szegö inequality for defined function class. We also discuss the same problems for the inverses of functions in this subclass. Further, we give a few corollaries for particular parameter values. (English) |
| Keyword:
|
generalized bivariate Fibonacci polynomial |
| Keyword:
|
analytic function |
| Keyword:
|
bi-univalent function |
| Keyword:
|
coefficient estimate |
| Keyword:
|
Fekete-Szegö functional |
| Keyword:
|
coefficients of inverse function |
| MSC:
|
30C45 |
| MSC:
|
33C50 |
| DOI:
|
10.21136/MB.2025.0063-25 |
| . |
| Date available:
|
2026-09-23T08:24:20Z |
| Last updated:
|
2026-09-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153769 |
| . |
| Reference:
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[1] Al-Asadi, A. M. A., Al-Ziadi, N. A. J.: Exploring new subclasses of bi-univalent functions using Gegenbauer polynomials: Coefficient bounds and Fekete-Szegö problems.AIP Conf. Proc. 3169 (2025), Article ID 070013. 10.1063/5.0254958 |
| Reference:
|
[2] nkaya, Ş. Altı, n, S. Yalçı: Faber polynomial coefficient bounds for a subclass of bi-univalent functions.C. R., Math., Acad. Sci. Paris 353 (2015), 1075-1080. Zbl 1335.30005, MR 3427911, 10.1016/j.crma.2015.09.003 |
| Reference:
|
[3] nkaya, Ş. Altı, n, S. Yalçı: On the $(p,q)$-Lucas polynomial coefficient bounds of the bi-univalent function class $\sigma$.Bol. Soc. Mat. Mex., III. Ser. 25 (2019), 567-575. Zbl 1430.30006, MR 4022306, 10.1007/s40590-018-0212-z |
| Reference:
|
[4] nkaya, Ş. Altı, n, S. Yalçı: On the Chebyshev polynomial coefficient problem of bi-Bazilevič function.TWMS J. Appl. Eng. Math. 10 (2020), 251-258. |
| Reference:
|
[5] Amourah, A.: Initial bounds for analytic and bi-univalent functions by means of $(p,q)$-Chebyshev polynomials defined by differential operator.Gen. Lett. Math. 7 (2019), 45-51. 10.31559/glm2019.7.2.1 |
| Reference:
|
[6] Amourah, A.: Fekete-Szegö inequality for analytic and bi-univalent functions subordinate to $(p,q)$-Lucas polynomials.TWMS J. Appl. Eng. Math. 11 (2021), 959-965. |
| Reference:
|
[7] Amourah, A., Frasin, B. A., Murugusundaramoorthy, G., Al-Hawary, T.: Bi-Bazilevič functions of order $ \nu+i\delta$ associated with $(p,q)$-Lucas polynomials.AIMS Math. 6 (2021), 4296-4305. Zbl 1484.30005, MR 4220409, 10.3934/math.2021254 |
| Reference:
|
[8] Brannan, D. A., (Eds.), J. G. Clunie: Aspects of Contemporary Complex Analysis.Academic Press, New York (1980). Zbl 0483.00007, MR 0623462 |
| Reference:
|
[9] Brannan, D. A., Taha, T. S.: On some classes of bi-univalent functions.Mathematical Analysis and its Applications KFAS Proceedings Series 3. Pergamon Press, Oxford (1988), 53-60. Zbl 0703.30014, MR 0951657, 10.1016/B978-0-08-031636-9.50012-7 |
| Reference:
|
[10] Bulut, S., Magesh, N., Balaji, V. K.: Initial bounds for analytic and bi-univalent functions by means of Chebyshev polynomials.J. Class. Anal. 11 (2017), 83-89. Zbl 1424.30033, MR 3671180, 10.7153/jca-11-06 |
| Reference:
|
[11] Das, R. N., Singh, P.: On subclasses of schlicht mappings.Indian J. Pure Appl. Math. 8 (1977), 864-872. Zbl 0374.30008, MR 0486470 |
| Reference:
|
[12] Duren, P. L.: Univalent Functions.Grundlehren der Mathematischen Wissenschaften 259. Springer, New York (1983). Zbl 0514.30001, MR 0708494 |
| Reference:
|
[13] Frasin, B. A., Swamy, S. R., Aldawish, I.: A comprehensive family of biunivalent functions defined by $k$-Fibonacci numbers.J. Funct. Spaces 2021 (2021), Article ID 4249509, 7 pages. Zbl 1482.30034, MR 4335587, 10.1155/2021/4249509 |
| Reference:
|
[14] Frasin, B. A., Swamy, S. R., Nirmala, J.: Some special families of holomorphic and Al-Oboudi type bi-univalent functions related to $k$-Fibonacci numbers involving modified Sigmoid activation function.Afr. Mat. 32 (2021), 631-643. Zbl 1488.30069, MR 4259360, 10.1007/s13370-020-00850-w |
| Reference:
|
[15] Gbolagade, A. M., Awolere, I. T., Adeyemo, O., Oladipo, A. T.: Application of the neutrosophic Poisson distribution series on the harmonic subclass of analytic functions using the Salagean derivative operator.Neutrosophic Syst. Appl. 23 (2024), 33-46. 10.61356/j.nswa.2024.23390 |
| Reference:
|
[16] Güney, H. Ã ., Murugusundaramoorthy, G., Sokół, J.: Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers.Acta Univ. Sapientiae, Math. 10 (2018), 70-84. Zbl 1412.30043, MR 3854091, 10.2478/ausm-2018-0006 |
| Reference:
|
[17] Güney, H. Ã ., Murugusundaramoorthy, G., Sokół, J.: Certain subclasses of bi-univalent functions related to $k$-Fibonacci numbers.Commun. Fac. Sci. Univ. Ank., Sér. A1., Math. Stat. 68 (2019), 1909-1921. Zbl 1494.30026, MR 4010330, 10.31801/cfsuasmas.505287 |
| Reference:
|
[18] Hamidi, S. G., Jahangiri, J. M.: Faber polynomial coefficient estimates for bi-univalent functions defined by subordinations.Bull. Iran. Math. Soc. 41 (2015), 1103-1119. Zbl 1373.30019, MR 3416618 |
| Reference:
|
[19] Hohlov, J. E.: Operators and operations on the class of univalent functions.Izv. Vyssh. Uchebn. Zaved., Mat. 10 (1978), 83-89 Russian. Zbl 0406.30009, MR 0522766 |
| Reference:
|
[20] Hu, Q., Shaba, T. G., Younis, J., Khan, B., Mashwani, W. K., Çağlar, M.: Applications of $q$-derivative operator to subclasses of bi-univalent functions involving Gegenbauer polynomials.Appl. Math. Sci. Eng. 30 (2022), 501-520. MR 4442844, 10.1080/27690911.2022.2088743 |
| Reference:
|
[21] Kocer, E. G., Tuncez, S.: Bivariate Fibonacci and Lucas like polynomials.Gazi Univ. J. Sci. 29 (2016), 109-113. |
| Reference:
|
[22] 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:
|
[23] Miller, S. S., Mocanu, P. T.: Differential Subordinations: Theory and Applications.Pure and Applied Mathematics, Marcel Dekker 225. Marcel Dekker, New York (2000). Zbl 0954.34003, MR 1760285, 10.1201/9781482289817 |
| Reference:
|
[24] Netanyahu, E.: The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $|z|<1$.Arch. Ration. Mech. Anal. 32 (1969), 100-112. Zbl 0186.39703, MR 0235110, 10.1007/BF00247676 |
| Reference:
|
[25] Noor, K. I., Noor, M. A.: On integral operators.J. Math. Anal. Appl. 238 (1999), 341-352. Zbl 0934.30007, MR 1715487, 10.1006/jmaa.1999.6501 |
| Reference:
|
[26] Orhan, H., Aktaş, İ., kan, H. Arı: On new subclasses of biunivalent functions associated with the $(p,q)$-Lucas polynomials and bi-Bazilevič type functions of order $\rho+i\xi$.Turk. J. Math. 47 (2023), 98-109. Zbl 1506.30005, MR 4556516, 10.55730/1300-0098.3348 |
| Reference:
|
[27] Sakaguchi, K.: On a certain univalent mapping.J. Math. Soc. Japan 11 (1959), 72-75. Zbl 0085.29602, MR 0107005, 10.2969/jmsj/01110072 |
| Reference:
|
[28] Sălăgean, G. S.: Subclasses of univalent functions.Complex Analysis: Fifth Romanian-Finnish Seminar. Part 1 Lecture Notes in Mathematics 1013. Springer, Berlin (1983), 362-372. Zbl 0531.30009, MR 0738107, 10.1007/BFb0066543 |
| Reference:
|
[29] Saravanan, G., Baskaran, S., Vanithakumari, B., Bulut, S.: Sakaguchi type functions defined by balancing polynomials.Math. Bohem. 150 (2025), 71-83. Zbl 08063798, MR 4869943, 10.21136/MB.2024.0173-23 |
| Reference:
|
[30] Srivastava, H. M., nkaya, Ş. Altı, n, S. Yalçı: Certain subclasses of bi-univalent functions associated with the Horadam polynomials.Iran. J. Sci. Technol., Trans. A Sci. 43 (2019), 1873-1879. MR 3978601, 10.1007/s40995-018-0647-0 |
| Reference:
|
[31] 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:
|
[32] 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:
|
[33] Srivastava, H. M., Murugusundaramoorthy, G., Magesh, N.: Certain subclasses of bi-univalent functions associated with the Hohlov operator.Global J. Math. Anal. 1 (2013), 67-73. |
| Reference:
|
[34] Srivastava, H. M., Wanas, A. K., Murugusundaramoorthy, G.: A certain family of bi-univalent functions associated with the Pascal distribution series based upon the Horadam polynomials.Surv. Math. Appl. 16 (2021), 193-205. Zbl 1470.30017, MR 4224411 |
| Reference:
|
[35] Srivastava, H. M., Wanas, A. K., Srivastava, R.: Applications of the $q$-Srivastava-Attiya operator involving a certain family of bi-univalent functions associated with the Horadam polynomials.Symmetry 13 (2021), Article ID 1230, 14 pages. 10.3390/sym13071230 |
| Reference:
|
[36] Wanas, A. K., Swamy, S. R.: Applications of the second kind Chebyshev polynomials of bi-starlike and bi-convex $\lambda$-pseudo functions associated with Sakaguchi type functions.Earthline J. Math. Sci. 13 (2023), 497-507. 10.34198/ejms.13223.497507 |
| Reference:
|
[37] Yilmaz, N., Aktaş, İ.: On some new subclasses of bi-univalent functions defined by generalized bivariate Fibonacci polynomial.Afr. Mat. 33 (2022), Article ID 59, 12 pages. Zbl 1542.30012, MR 4417363, 10.1007/s13370-022-00993-y |
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