Applications of Horadam Polynomials for Bazilevič and λ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi Type Functions
Abstract
:1. Introduction and Preliminaries
- 1.
- Considering , we derive the Fibonacci polynomials .
- 2.
- Considering and , we obtain the Lucas polynomials .
- 3.
- Considering and , we get the Pell polynomials .
- 4.
- Considering and , we have the Pell-Lucas polynomials .
- 5.
- Considering , and , we find the Chebyshev polynomials of the first kind.
- 6.
- Considering , and , we deduce the Chebyshev polynomials of the second kind.
2. Main Results
- 1.
- Considering , we get
- 2.
- Taking , we reobtain
- 3.
- 4.
- For conditions , , , and , we find
- 5.
- For , , , and , we have
- 6.
- For , , , and , we deduce
- 7.
- For condition , , , and , we derived
- 8.
- Second item;
- 9.
- Considering , , , , and , , we have
- 10.
- Taking , , , , and , , we have
- 11.
- For , , , , and , , we have
- 12.
- Considering , , , , and , , we have
3. Special Cases and Consequences
- 1.
- If we let in our Theorems, we have the results for well-known class of bi-Bazilevič functions which was recently investigated by Wanas and Lupas [31].
- 2.
- Considering in our Theorems, we obtain the results for the class which was obtained recently by Srivastava et al. [29].
- 3.
- Taking and in our Theorems, we get the results for the well-known class which was studied recently by Alamoush [30].
- 4.
- If we let , , , and in our Theorems, one can obtains the results for the class of bi-Bazilevič functions which was discussed recently by Bulut et al. [36].
- 5.
- Considering , , , and in our Theorems, we derive the results for the family of bi-pseudo-starlike functions which was recently investigated by Magesh and Bulut [37].
- 6.
- If we put , , , and in our Theorems, we obtain the results for the family of bi-starlike functions which was considered recently by Altınkaya and Yalçin [38].
- 7.
- If we let , , , and in our Theorems, we find the results for the class which was discussed recently by Bulut et al. [36].
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Al-Shbeil, I.; Wanas, A.K.; AlAqad, H.; Cătaş, A.; Alohali, H. Applications of Horadam Polynomials for Bazilevič and λ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi Type Functions. Symmetry 2024, 16, 218. https://doi.org/10.3390/sym16020218
Al-Shbeil I, Wanas AK, AlAqad H, Cătaş A, Alohali H. Applications of Horadam Polynomials for Bazilevič and λ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi Type Functions. Symmetry. 2024; 16(2):218. https://doi.org/10.3390/sym16020218
Chicago/Turabian StyleAl-Shbeil, Isra, Abbas Kareem Wanas, Hala AlAqad, Adriana Cătaş, and Hanan Alohali. 2024. "Applications of Horadam Polynomials for Bazilevič and λ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi Type Functions" Symmetry 16, no. 2: 218. https://doi.org/10.3390/sym16020218
APA StyleAl-Shbeil, I., Wanas, A. K., AlAqad, H., Cătaş, A., & Alohali, H. (2024). Applications of Horadam Polynomials for Bazilevič and λ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi Type Functions. Symmetry, 16(2), 218. https://doi.org/10.3390/sym16020218