Gegenbauer Polynomials for a Subfamily of Bi-univalent Functions

  • Sondekola Rudra Swamy Department of Information Science and Engineering, Acharya Institute of Technology, Bengaluru-560 107, Karnataka, India
  • Manje Gowda Kavana Department of Information Science and Engineering, Acharya Institute of Technology, Bengaluru-560 107, Karnataka, India
  • Paniraj Nagashree Department of Information Science and Engineering, Acharya Institute of Technology, Bengaluru-560 107, Karnataka, India
  • Mali Triveni Department of Information Science and Engineering, Acharya Institute of Technology, Bengaluru-560 107, Karnataka, India
  • Sanaulla Irfan Department of Information Science and Engineering, Acharya Institute of Technology, Bengaluru-560 107, Karnataka, India
DOI: https://doi.org/10.34198/ejms.15425.461471
Keywords: regular functions, subordination, bi-univalent functions, Gegenbauer polynomials

Abstract

We study a subfamily of bi-univalent and regular functions in the open unit disk subordinate to Gegenbauer polynomials. For functions in the defined subfamily, we derive initial coefficients bounds. Additionally, the Fekete-Szegö problem is handled for the elements of the defined subfamily. We also discuss relevant connections to previous findings and several fresh outcomes are shown to follow.

References

Amourah, A., Alamoush, A., & Al-Kaseasbeh, M. (2021). Gegenbauer polynomials and biunivalent functions. Palestine Journal of Mathematics, 10(2), 625-632. https://doi.org/10.1155/2021/5574673

Bieberbach, L. (1916). Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. Sitzungsberichte der Preußischen Akademie der Wissenschaften, 138, 940-955.

Brannan, D.A., & Clunie, J.G. (1979). Aspects of contemporary complex analysis. In Proceedings of the NATO Advanced Study Institute held at University of Durham. New York: Academic Press.

Brannan, D.A., & Taha, T.S. (1986). On some classes of bi-univalent functions. Studia Universitatis Babeș-Bolyai Mathematica, 31(2), 70-77.

Bulut, S. (2014). Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions. Comptes Rendus de l'Académie des Sciences, Série I, 352, 479-484. https://doi.org/10.1016/j.crma.2014.04.004

Çağlar, M., Deniz, E., & Srivastava, H.M. (2017). Second Hankel determinant for certain subclasses of bi-univalent functions. Turkish Journal of Mathematics, 41, 694-706. https://doi.org/10.3906/mat-1602-25

De Branges, L. (1985). A proof of the Bieberbach conjecture. Acta Mathematica, 154(1-2), 137-152. https://doi.org/10.1007/BF02392821

Deniz, E. (2013). Certain subclasses of bi-univalent functions satisfying subordinate conditions. Journal of Classical Analysis, 2(1), 49-60. https://doi.org/10.7153/jca-02-05

Duren, P.L. (1983). Univalent Functions. Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York.

Frasin, B.A. (2014). Coefficient bounds for certain classes of bi-univalent functions. Hacettepe Journal of Mathematics and Statistics, 43(3), 383-389.

Frasin, B.A., & Aouf, M.K. (2011). New subclasses of bi-univalent functions. Applied Mathematics, 24, 1569-1573. https://doi.org/10.1016/j.aml.2011.03.048

Frasin, B.A., Swamy, S.R., & Aldawish, I. (2021). A comprehensive family of bi-univalent functions defined by k-Fibonacci numbers. Journal of Function Spaces, 2021(1), 4249509. https://doi.org/10.1155/2021/4249509

Frasin, B.A., Swamy, S.R., Amourah, A., Salah, J., & Maheshwarappa, R.H. (2024). A family of bi-univalent functions defined by (p, q)-derivative operator subordinate to a generalized bivariate Fibonacci polynomials. European Journal of Pure and Applied Mathematics, 17(4), 3801-3814. https://doi.org/10.29020/nybg.ejpam.v17i4.5526

Fekete, M., & Szegö, G. (1933). Eine Bemerkung über ungerade schlichte Funktionen. Journal of the London Mathematical Society, 89, 85-89. https://doi.org/10.1112/jlms/s1-8.2.85

Guney, H.Ö., Murugusundaramoorthy, G., & Sokol, J. (2018). Certain subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers. Acta Universitatis Sapientiae Mathematica, 10, 70-84. https://doi.org/10.2478/ausm-2018-0006

Kiepiela, K., Naraniecka, I., & Szynal, J. (2003). The Gegenbauer polynomials and typically real functions. Journal of Computational and Applied Mathematics, 153(1-2), 273-282. https://doi.org/10.1016/S0377-0427(02)00642-8

Lewin, M. (1967). On a coefficient problem for bi-univalent functions. Proceedings of the American Mathematical Society, 18, 63-68. https://doi.org/10.1090/S0002-9939-1967-0206255-1

Netenyahu, E. (1969). The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|<1. Archive for Rational Mechanics and Analysis, 32, 100-112. https://doi.org/10.1007/BF00247676

Orhan, H., Mamatha, P. K., Swamy, S.R., Magesh, N., & Yamini, J. (2021). Certain classes of bi-univalent functions associated with the Horadam polynomials. Acta Universitatis Sapientiae, Mathematica, 13(1), 258-272. https://doi.org/10.2478/ausm-2021-0015

Ozturk, R., & Aktas, I. (2024). Coefficient estimate and Fekete-Szego problems for certain new subclasses of bi-univalent functions defined by generalized bivariate Fibonacci polynomial. Sahand Communications in Mathematical Analysis, 21(3), 35-53. https://doi.org/10.22130/scma.2023.1988826.124

Saravanan, G., Baskaran, S., Vanithakumari, B., Alnaji, L., Shaba, T.M., Al-Shbeil, I., & Lupas, A.A. (2024). Bernoulli polynomials for a new subclass of Te-univalent functions. Heliyon, 10, e33953, 1-11. https://doi.org/10.1016/j.heliyon.2024.e33953

Shammaky, A.E., Frasin, B.A., & Swamy, S.R. (2022). Fekete-Szego inequality for bi-univalent functions subordinate to Horadam polynomials. Journal of Functional Spaces, 2022(1), 9422945. https://doi.org/10.1155/2022/9422945

Srivastava, H.M., Mishra, A.K., & Gochhayat, P. (2010). Certain subclasses of analytic and bi-univalent functions. Applied Mathematics Letters, 23, 1188-1192. https://doi.org/10.1016/j.aml.2010.05.009

Srivastava, H.M., Altinkaya, S., & Yalcin, S. (2019). Certain subclasses of bi-univalent functions associated with the Horadam polynomials. Iranian Journal of Science and Technology, Transactions of Sciences, 43, 1873-1879. https://doi.org/10.1007/s40995-018-0647-0

Swamy, S.R. (2023). Bi-univalent function subclasses subordinate to Horadam polynomials. Earthline Journal of Mathematical Sciences, 11(2), 183-198. https://doi.org/10.34198/ejms.11223.183198

Swamy, S.R., & Atshan, W.G. (2022). Coefficient bounds for regular and bi-univalent functions linked with Gegenbauer polynomials. Gulf Journal of Mathematics, 13(2), 67-77. https://doi.org/10.56947/gjom.v13i2.723

Swamy, S.R., Breaz, D., Kala, V., Mamatha, P.K., Cotirla, L.-I., & Rapeanu, E. (2024). Initial coefficient bounds analysis for novel subclasses of bi-univalent functions linked with Lucas-Balancing polynomials. Mathematics, 12, 1325. https://doi.org/10.3390/math12091325

Swamy, S.R., Frasin, B.A., Breaz, D., & Cotirla, L.-I. (2024). Two families of bi-univalent functions associating the (p, q)-derivative with generalized bivariate Fibonacci polynomials. Mathematics, 12, 3933. https://doi.org/10.3390/math12243933

Swamy, S.R., Nanjadeva, Y., Kumar, P., & Sushma, T.M. (2024). Initial coefficient bounds analysis for novel subclasses of bi-univalent functions linked with Horadam polynomials. Earthline Journal of Mathematical Sciences, 14(3), 443-457. https://doi.org/10.34198/ejms.14324.443457

Swamy, S.R., & Yalcin, S. (2022). Coefficient bounds for regular and bi-univalent functions linked with Gegenbauer polynomials. Problems of Analysis: Issues of Analysis, 11(1), 133-144. https://doi.org/10.15393/j3.art.2022.10351

Swamy, S.R., Yalcin, S., Mamatha, P.K., Chauhan, G.S., & Jagadish, N. (2025). Certain subclasses of bi-univalent functions governed by generalized bivariate Fibonacci polynomials. Advances in Nonlinear Variational Inequalities, 28(6s), 119-130.

Swamy, S.R., & Wanas, A.K. (2022). A comprehensive family of bi-univalent functions defined by (m, n)-Lucas polynomials. Boletin de la Sociedad Matematica Mexicana, 28, 34. https://doi.org/10.1007/s40590-022-00411-0

Tan, D.L. (1984). Coefficient estimates for bi-univalent functions. Chinese Annals of Mathematics, Series A, 5, 559-568.

Tang, H., Deng, G., & Li, S. (2013). Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions. Journal of Inequalities and Applications, 2013, Article 317, 10 pages. https://doi.org/10.1186/1029-242X-2013-317

Wanas, A.K., Swamy, S.R., Tang, H., Shaba, T.G., Nirmala, J., & Ibrahim, I.O. (2021). A comprehensive family of bi-univalent functions linked with Gegenbauer polynomials. Turkish Journal of Inequalities, 5(2), 61-69.

Wanas, A.K., Frasin, B.A., Swamy, S.R., & Sailaja, Y. (2022). Coefficient bounds for a family of bi-univalent functions defined by Horadam polynomials. Acta et Commentationes Universitatis Tartuensis de Mathematica, 26(1), 25-32. https://doi.org/10.12697/ACUTM.2022.26.02

Published
2025-04-02
How to Cite
Swamy, S. R., Kavana, M. G., Nagashree, P., Triveni, M., & Irfan, S. (2025). Gegenbauer Polynomials for a Subfamily of Bi-univalent Functions. Earthline Journal of Mathematical Sciences, 15(4), 461-471. https://doi.org/10.34198/ejms.15425.461471
Issue
Vol 15 No 4 (2025): In progress
Section
Articles


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