$q$--Rabotnov Functions and Bi-univalent Functions

Authors

  • Abdullah Alsoboh College of Applied and Health Sciences, ASharqiyah University, Post Box No. 42,\\ Post Code No. 400, Ibra, Sultanate of Oman
  • Ala Amourah Department of Mathematics, Faculty of Education and Arts, Sohar University, Sohar 3111, Sultanate of Oman,
  • Omar Alnajar Department of Mathematical Sciences, Faculty of Science and Technology, \\ Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia
  • Fahad Al Abri College of Applied and Health Sciences, ASharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Sultanate of Oman
  • Tala Sasa Department of Mathematics, Faculty of Science, Applied Science Private University, Amman, Jordan

DOI:

https://doi.org/10.29020/nybg.ejpam.v19i1.7109

Keywords:

Subordination, ultraspherical Polynomial, $\q$-Calculus, analytic functions,, univalent functions, bi-univalent functions, orthogonal polynomials, carathéodory's functions, fekete-Szeg\"{o} problem

Abstract

In this work, we introduce and investigate a new subclass of bi-univalent functions, denoted by $\mathfrak{R}_{\Sigma}^{\mu,x}(\varrho,\varphi,\lambda;\gamma;q)$, which is defined through the interplay between $q$--Rabotnov functions and $q$--Gegenbauer polynomials. For functions in this class, we derive sharp bounds for the initial Taylor--Maclaurin coefficients $|a_{2}|$ and $|a_{3}|$, and we establish estimates for the corresponding Fekete--Szeg\H{o} functional. Furthermore, by selecting suitable parameter values, our results reduce to several well known subclasses, thereby yielding a range of new consequences and extensions in the theory of analytic bi-univalent functions.

Author Biography

  • Fahad Al Abri, College of Applied and Health Sciences, ASharqiyah University, Post Box No. 42, Post Code No. 400, Ibra, Sultanate of Oman

    College of Applied and Health Sciences, ASharqiyah University, Post Box No. 42,
    Post Code No. 400, Ibra, Sultanate of Oman

References

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Published

2026-02-22

Issue

Vol. 19 No. 1: (January 2026)

Section

Complex Analysis

License

Copyright (c) 2026 Abdullah Alsoboh, Ala Amourah, Omar Alnajar, Fahad Al Abri, Tala Sasa

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How to Cite

$q$--Rabotnov Functions and Bi-univalent Functions. (2026). European Journal of Pure and Applied Mathematics, 19(1), 7109. https://doi.org/10.29020/nybg.ejpam.v19i1.7109