Bi-univalent functions connected to Bazilevič and $ \lambda $-Pseudo functions and their Lucas-Balancing polynomial applications

Abbas Kareem Wanas; H. M. Srivastava; Adriana Cătaş; Sheza M. El-Deeb; Abbas Kareem Wanas; H. M. Srivastava; Adriana Cătaş; Sheza M. El-Deeb

doi:10.3934/math.20251115

AIMS Mathematics

2025, Volume 10, Issue 11: 25193-25205. doi: 10.3934/math.20251115

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Bi-univalent functions connected to Bazilevič and $ \lambda $-Pseudo functions and their Lucas-Balancing polynomial applications

1.
Department of Mathematics, College of Education for Women, University of Al-Qadisiyah, Al Diwaniyah 58001, Al-Qadisiyah, Iraq
2.
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W $ 3R4 $, Canada
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4.
Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea
5.
Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan City 320314, Taiwan
6.
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku $ AZ1007 $, Azerbaijan
7.
Section of Mathematics, International Telematic University Uninettuno, 39 Corso Vittorio Emanuele Ⅱ, Ⅰ-00186 Rome, Italy
8.
Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania
9.
Department of Mathematics, College of Science, Qassim University, Buraydah, 51452, Saudi Arabia

Received: 23 April 2025 Revised: 18 July 2025 Accepted: 22 July 2025 Published: 03 November 2025
MSC : 30C20, 30C45

In this work, we define two families $ \mathcal{V}_{\Sigma }(\mu, \gamma, \lambda; r) $ and $ \mathcal{W}_{\Sigma }(\mu, \gamma, \lambda; r) $ of holomorphic and bi-univalent functions connected with Bazilevič functions and $ \lambda $-pseudo functions defined by Lucas-Balancing polynomials. We demonstrate the upper bounds for the initial Taylor-Maclaurin coefficients. In addition, the Fekete-Szegö type inequalities are derived for functions in these families. Moreover, we indicate certain special cases and consequences for our results.
- bi-univalent function,
- holormorphic function,
- Bazilevič function,
- $ \lambda $-Pseudo functions,
- upper bounds,
- Lucas-Balancing polynomials,
- convolution, Fekete-Szegö problem
Citation: Abbas Kareem Wanas, H. M. Srivastava, Adriana Cătaş, Sheza M. El-Deeb. Bi-univalent functions connected to Bazilevič and $ \lambda $-Pseudo functions and their Lucas-Balancing polynomial applications[J]. AIMS Mathematics, 2025, 10(11): 25193-25205. doi: 10.3934/math.20251115

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Abstract

In this work, we define two families $ \mathcal{V}_{\Sigma }(\mu, \gamma, \lambda; r) $ and $ \mathcal{W}_{\Sigma }(\mu, \gamma, \lambda; r) $ of holomorphic and bi-univalent functions connected with Bazilevič functions and $ \lambda $-pseudo functions defined by Lucas-Balancing polynomials. We demonstrate the upper bounds for the initial Taylor-Maclaurin coefficients. In addition, the Fekete-Szegö type inequalities are derived for functions in these families. Moreover, we indicate certain special cases and consequences for our results.

References

[1]	A. Behera, G. K. Panda, On the square roots of triangular numbers, Fibonacci Quart., 37 (1999), 98–105. https://doi.org/10.1080/00150517.1999.12428864 doi: 10.1080/00150517.1999.12428864
[2]	S. Z. H. Bukhari, A. K. Wanas, M. Abdalla, S. Zafar, Region of variablity for Bazilevič functions, AIMS Mathematics, 8 (2023), 25511–25527. https://doi.org/10.3934/math.20231302 doi: 10.3934/math.20231302
[3]	L. I. Cotîrlǎ, New classes of analytic and bi-univalent functions, AIMS Mathematics, 6 (2021), 10642–10651. https://doi.org/10.3934/math.2021618 doi: 10.3934/math.2021618
[4]	L. I.Cotîrlǎ, A. K. Wanas, Applications of Laguerre polynomials for Bazilevič and $\theta $-pseudo-starlike bi-univalent functions associated with Sakaguchi-type functions, Symmetry, 15 (2023), 406. https://doi.org/10.3390/sym15020406 doi: 10.3390/sym15020406
[5]	P. L. Duren, Univalent functions, In: Grundlehren der mathematischen Wissenschaften, New York: Springer-Verlag, 259 (1983).
[6]	S. M. El-Deeb, T. Bulboacǎ, B. M. El-Matary, Maclaurin coefficient estimates of bi-univalent functions connected with the $q$-derivative, Mathematics, 8 (2020), 418. https://doi.org/10.3390/math8030418 doi: 10.3390/math8030418
[7]	M. Fekete, G. Szegö, Eine bemerkung uber ungerade schlichte funktionen, J. London Math. Soc., s1-8 (1933), 85–89. https://doi.org/10.1112/jlms/s1-8.2.85 doi: 10.1112/jlms/s1-8.2.85
[8]	R. Frontczak, A note on hybrid convolutions involving balancing and Lucas-Balancing numbers, Appl. Math. Sci., 12 (2018), 1201–1208. https://doi.org/10.12988/ams.2018.87111 doi: 10.12988/ams.2018.87111
[9]	R. Frontczak, On balancing polynomials, Appl. Math. Sci., 13 (2019), 57–66. https://doi.org/10.12988/ams.2019.812183 doi: 10.12988/ams.2019.812183
[10]	R. Keskin, O. Karaatli, Some new properties of balancing numbers and square triangular numbers, J. Integer Seq., 15 (2012), 1–13.
[11]	S. L. Krushkal, A variational theory for biunivalent holomorphic functions, Axioms, 13 (2024), 628. https://doi.org/10.3390/axioms13090628 doi: 10.3390/axioms13090628
[12]	S. L. Krushkal, Teichmüller balls and biunivalent holomorphic functions, J. Math. Sci., 288 (2025), 225–233. https://doi.org/10.1007/s10958-025-07678-4 doi: 10.1007/s10958-025-07678-4
[13]	G. Murugusundaramoorthy, L. I. Cotîrlǎ, D. Breaz, S. M. El-Deeb, Applications of Lucas Balancing polynomial to subclasses of bi-starlike functions, Axioms, 14 (2025), 50. https://doi.org/10.3390/axioms14010050 doi: 10.3390/axioms14010050
[14]	R. Öztürk, I. Aktaş, Coefficient estimates for two new subclasses of bi-univalent functions defined by Lucas-Balancing polynomials, Turkish J. Ineq., 7 (2023), 55–64.
[15]	B. K. Patel, N. Irmak, P. K. Ray, Incomplete balancing and Lucas-balancing numbers, Math. Rep., 20 (2018), 59–72.
[16]	P. K. Ray, Some congruences for balancing and Lucas-Balancing numbers and their applications, 2014.
[17]	R. Singh, On Bazilevič functions, Proc. Amer. Math. Soc., 38 (1973), 261–271. https://doi.org/10.1090/S0002-9939-1973-0311887-9 doi: 10.1090/S0002-9939-1973-0311887-9
[18]	H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188–1192. https://doi.org/10.1016/j.aml.2010.05.009 doi: 10.1016/j.aml.2010.05.009
[19]	H. M. Srivastava, A. K. Wanas, Applications of the Horadam polynomials involving $\lambda$-pseudo-starlike bi-univalent functions associated with a certain convolution operator, Filomat, 35 (2021), 4645–4655. https://doi.org/10.2298/FIL2114645S doi: 10.2298/FIL2114645S
[20]	A. K. Wanas, Coefficient estimates for Bazilevič functions of bi-prestarlike functions, Miskolc Math. Notes, 21 (2020), 1031–1040. https://doi.org/10.18514/MMN.2020.3174 doi: 10.18514/MMN.2020.3174

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