Certain subfamily of Bazilevič holomorphic functions involving the $ m $-leaf polynomial

Khalid M. K. Alshammari; Tamer M. Seoudy; Amr K. Amin; Khalid M. K. Alshammari; Tamer M. Seoudy; Amr K. Amin

doi:10.3934/math.2026644

AIMS Mathematics

2026, Volume 11, Issue 6: 15676-15690. doi: 10.3934/math.2026644

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Certain subfamily of Bazilevič holomorphic functions involving the $ m $-leaf polynomial

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia; khmo.alshammari@uoh.edu.sa
2.
Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt; tms00@fayoum.edu.eg
3.
Department of Mathematics, Adham University College, Umm AL-Qura University, Makkah, Saudi Arabia; akgadelrab@uqu.edu.sa

Received: 18 February 2026 Revised: 16 May 2026 Accepted: 29 May 2026 Published: 04 June 2026
MSC : 30C45

In this paper, we introduce and investigate a novel subfamily of Bazilevič functions, defined using the $ m $-leaf polynomial and the principle of subordination. For this subfamily, we establish several properties, including subordination results, inclusion relationships, and sharp coefficient bounds. Furthermore, we derive a general Fekete–Szegö inequality, providing a solution to this classical problem for functions within the new subfamily.
- Holomorphic functions,
- subordination,
- Bazilevičfunction,
- $ m $-Leaf function,
- Fekete–Szegöproblem
Citation: Khalid M. K. Alshammari, Tamer M. Seoudy, Amr K. Amin. Certain subfamily of Bazilevič holomorphic functions involving the $ m $-leaf polynomial[J]. AIMS Mathematics, 2026, 11(6): 15676-15690. doi: 10.3934/math.2026644

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Abstract

In this paper, we introduce and investigate a novel subfamily of Bazilevič functions, defined using the $ m $-leaf polynomial and the principle of subordination. For this subfamily, we establish several properties, including subordination results, inclusion relationships, and sharp coefficient bounds. Furthermore, we derive a general Fekete–Szegö inequality, providing a solution to this classical problem for functions within the new subfamily.

References

[1]	A. S. Alshehry, R. Shah, A. Bariq, The second Hankel determinant of logarithmic coefficients for starlike and convex functions involving four-leaf-shaped domain, J. Funct. Space., 2022 (2022), 2621811. http://dx.doi.org/10.1155/2022/2621811 doi: 10.1155/2022/2621811
[2]	I. E. Bazilevič, On a case of integrability in quadratures of the Loewner–Kufarev equation, Matematicheskii Sbornik, 37 (1955), 471–476.
[3]	T. Bulboacă, Differential subordinations and superordinations: Recent results, House of Scientific Book Publ., Cluj-Napoca, 2005.
[4]	J. H. Choi, M. Saigo, H. M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl., 276 (2002), 432–445. http://dx.doi.org/10.1016/s0022-247x(02)00500-0 doi: 10.1016/s0022-247x(02)00500-0
[5]	M. Fekete, G. Szegö, Eine Bemerkung Über ungerade schlichte Funktionen, J. Lond. Math. Soc., 8 (1933), 85–89. http://dx.doi.org/10.1112/jlms/s1-8.2.85 doi: 10.1112/jlms/s1-8.2.85
[6]	S. Gandhi, Radius estimates for three leaf function and convex combination of starlike functions, In: Mathematical analysis I: Approximation theory, Springer Proceedings in Mathematics & Statistics, Singapore: Springer, 306 (2020), 173–184. https://doi.org/10.1007/978-981-15-1153-0_15
[7]	D. J. Hallenbeck, S. Ruschewyh, Subordination by convex functions, P. Am. Math. Soc., 52 (1975), 191–195. https://doi.org/10.1090/S0002-9939-1975-0374403-3 doi: 10.1090/S0002-9939-1975-0374403-3
[8]	W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Pol. Math., 23 (1970), 159–177.
[9]	H. Zhou, K. A. Selvakumaran, S. Sivasubramanian, S. D. Purohit, H. Tang, Subordination problems for a new class of Bazilevič functions associated with $k$-symmetric points and fractional $q$-calculus operators, AIMS Math., 6 (2021), 8642–8653. http://dx.doi.org/10.3934/math.2021502 doi: 10.3934/math.2021502
[10]	P. Kavitha, V. K. Balaji, T. Stalin, On the Fekete-Szegö inequality for analytic functions via Hohlov operator on leaf-like domains, Results Nonlinear Anal., 8 (2025), 172–183. http://dx.doi.org/10.31838/rna/2025.08.01.013 doi: 10.31838/rna/2025.08.01.013
[11]	V. S. Kumar, R. B. Sharma, M. Haripriya, Third Hankel determinant for Bazilevic functions related to a leaf like domain, In: The 11th National Conference on Mathematical Techniques and Applications, 2112 (2019), 020088.
[12]	M. S. Liu, On certain subclass of analytic functions, J. South China Normal Univ., 4 (2002), 15–20.
[13]	W. Ma, D. A. Minda, A unified treatment of some special classes of univalent functions, In: Proceedings of the Conference on Complex Analysis, USA: Int. Press Inc, 1992.
[14]	S. S. Miller, P. T. Mocanu, Differential subordinations: Theory and applications, CRC Press, 225 (2000).
[15]	C. Pommerenke, Univalent functions, Göttingen: Vandenhoeck & Ruprecht, 1975.
[16]	M. Raza, K. I. Noor, Subclass of Bazilevič functions of complex order, AIMS Math., 5 (2020), 2448–2460. http://dx.doi.org/10.3934/math.2020162 doi: 10.3934/math.2020162
[17]	M. S. Robertson, Certain classes of star-like functions, Mich. Math. J., 32 (1985), 135–140. http://dx.doi.org/10.1307/mmj/1029003181 doi: 10.1307/mmj/1029003181
[18]	T. M. Seoudy, A. E. Shammaky, On certain class of Bazilevič functions associated with the Lemniscate of Bernoulli, J. Funct. Space., 2020 (2020), 6622230. http://dx.doi.org/10.1155/2020/6622230 doi: 10.1155/2020/6622230
[19]	T. M. Seoudy, A. E. Shammaky, Certain subclass of multivalently Bazilevič and non-Bazilevič functions involving the Lemniscate of Bernoulli, Eur. J. Pure Appl. Math., 18 (2025), 5869. http://dx.doi.org/10.29020/nybg.ejpam.v18i2.5869 doi: 10.29020/nybg.ejpam.v18i2.5869
[20]	P. Sharma, S. Sivasubramanian, N. E. Cho, Initial coefficient bounds for certain new subclasses of bi-Bazilevič functions and exponentially bi-convex functions with bounded boundary rotation, Axioms, 13 (2024), 25. http://dx.doi.org/10.3390/axioms13010025 doi: 10.3390/axioms13010025
[21]	T. S. Small, On Bazilevič functions, Q. J. Math., 23 (1972), 135–142. http://dx.doi.org/10.1093/qmath/23.2.135 doi: 10.1093/qmath/23.2.135
[22]	H. M. Srivastava, S. Khan, S. N. Malik, F. Tchier, A. Saliu, Q. Xin, Faber polynomial coefficient inequalities for bi-Bazilevič functions associated with the Fibonacci-number series and the square-root functions, J. Inequal. Appl., 2024 (2024), 16. http://doi.org/10.1186/s13660-024-03090-9 doi: 10.1186/s13660-024-03090-9
[23]	H. M. Srivastava, T. G. Shaba, G. Murugusundaramoorthy, A. K. Wanas, G. I. Oros, The Fekete-Szegö functional and the Hankel determinant for a certain class of analytic functions involving the Hohlov operator, AIMS Math., 8 (2023), 340–360. http://dx.doi.org/10.3934/math.2023016 doi: 10.3934/math.2023016
[24]	B. Sudharsanan, S. Gunasekar, T. Bulboaca, Geometric properties of the $m$-Leaf function and connected subclasses, Symmetry, 17 (2025), 438. http://dx.doi.org/10.3390/sym17030438 doi: 10.3390/sym17030438
[25]	P. Sunthrayuth, Y. Jawarneh, M. Naeem, N. Iqbal, J. Kafle, Some sharp results on coefficient estimate problems for four-leaf-type bounded turning functions, J. Funct. Space., 2022 (2022), 8356125. http://dx.doi.org/10.1155/2022/8356125 doi: 10.1155/2022/8356125
[26]	S. Umar, M. Arif, M. Raza, S. K. Lee, On a subclass related to Bazilevič functions, AIMS Math., 5 (2020), 2040–2056. http://dx.doi.org/10.3934/math.2020135 doi: 10.3934/math.2020135
[27]	A. K. Wanas, S. A. Sehen, Á. O. P. Szabó, Toeplitz matrices for a class of Bazilevič functions and the $\lambda$-Pseudo-starlike functions, Axioms, 13 (2024), 521. http://dx.doi.org/10.3390/axioms13080521 doi: 10.3390/axioms13080521

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