Certain subclasses of analytic functions: Fekete-Szegö inequalities involving $ q $-Bessel functions and Carlson-Shaffer operator

Santosh Mandal; Madan Mohan Soren; K. R. Karthikeyan; Dharmaraj Mohankumar; Santosh Mandal; Madan Mohan Soren; K. R. Karthikeyan; Dharmaraj Mohankumar

doi:10.3934/math.2026930

AIMS Mathematics

2026, Volume 11, Issue 7: 23066-23089. doi: 10.3934/math.2026930

Previous Article Next Article

Research article Special Issues

Certain subclasses of analytic functions: Fekete-Szegö inequalities involving $ q $-Bessel functions and Carlson-Shaffer operator

1.
Department of Mathematics, Berhampur University, Berhampur, Odisha, 760007, India
2.
Department of Applied Mathematics and Science, National University of Science & Technology, Muscat, P.O. Box 620, Oman
3.
Department of Mathematics for Innovation, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences(SIMATS), Thandalam, Chennai, 602105, India

Received: 25 May 2026 Revised: 04 July 2026 Accepted: 10 July 2026 Published: 29 July 2026
MSC : 30C45, 30C80, 33E12

In this paper, we introduce a new linear operator involving the $ q $-analogue of generalized Bessel functions and the $ q $-analogue of the Carlson-Shaffer operator. By using this operator, we define and investigate certain subclasses of analytic functions in the open unit disk. We obtain coefficient estimates and establish Fekete-Szegö-type inequalities for the newly introduced function classes. Moreover, by employing the Hadamard product, we study several geometric properties and derive necessary as well as sufficient conditions for these classes. The results obtained in this work generalize and extend a number of earlier results in the existing literature.
- analytic functions,
- coefficient estimates,
- Fekete-Szegö inequalities,
- Hadamard product,
- Poisson distribution,
- $ q $-Bessel functions,
- $ q $-Carlson-Shaffer operator
Citation: Santosh Mandal, Madan Mohan Soren, K. R. Karthikeyan, Dharmaraj Mohankumar. Certain subclasses of analytic functions: Fekete-Szegö inequalities involving $ q $-Bessel functions and Carlson-Shaffer operator[J]. AIMS Mathematics, 2026, 11(7): 23066-23089. doi: 10.3934/math.2026930

Related Papers:

Abstract

In this paper, we introduce a new linear operator involving the $ q $-analogue of generalized Bessel functions and the $ q $-analogue of the Carlson-Shaffer operator. By using this operator, we define and investigate certain subclasses of analytic functions in the open unit disk. We obtain coefficient estimates and establish Fekete-Szegö-type inequalities for the newly introduced function classes. Moreover, by employing the Hadamard product, we study several geometric properties and derive necessary as well as sufficient conditions for these classes. The results obtained in this work generalize and extend a number of earlier results in the existing literature.

References

[1]	P. L. Duren, Univalent functions, New York: Springer, 1983.
[2]	M. Fekete, G. Szegö, Eine Bemerkung über ungerade schlichte Funktionen, J. Lond. Math. Soc., 8 (1933), 85–89.
[3]	F. R. Keogh, E. P. Merkes, A coefficient inequality for certain classes of analytic functions, P. Am. Math. Soc., 20 (1969), 8–12. https://doi.org/10.2307/2035949 doi: 10.2307/2035949
[4]	N. Alessa, B. Venkateswarlu, K. Loganathan, T. S. Karthik, P. T. Reddy, G. Sujatha, Certain class of analytic functions connected with $q$-analogue of the Bessel function, J. Math., 2021 (2021), 5587886. https://doi.org/10.1155/2021/5587886 doi: 10.1155/2021/5587886
[5]	S. M. El-Deeb, T. Bulboacǎ, Fekete–Szegö inequalities for certain class of analytic functions connected with $q$-analogue of Bessel function, J. Egypt. Math. Soc., 27 (2019), 42. https://doi.org/10.1186/s42787-019-0049-2 doi: 10.1186/s42787-019-0049-2
[6]	M. Mahmoud, Generalized $q$-Bessel function and its properties, Adv. Differ. Equ., 2013 (2013), 121. https://doi.org/10.1186/1687-1847-2013-121 doi: 10.1186/1687-1847-2013-121
[7]	G. Dattoli, A. Torre, Symmetric $q$-Bessel functions, Le Mat., 51 (1996), 153–167.
[8]	Q. Hu, R. S. Badar, M. G. Khan, Subclasses of $q$-uniformly starlike functions obtained via the $q$-Carlson–Shaffer operator, Axioms, 13 (2024), 842. https://doi.org/10.3390/axioms13120842 doi: 10.3390/axioms13120842
[9]	T. M. Seoudy, A. E. Shammaky, Certain subclasses of spiral-like functions associated with $q$-analogue of Carlson–Shaffer operator, AIMS Math., 6 (2020), 2525–2538. https://doi.org/10.3934/math.2021153 doi: 10.3934/math.2021153
[10]	G. Murugusundaramoorthy, Fekete–Szegö inequality for certain subclasses of analytic functions related with crescent-shaped domain and application of Poisson distribution series, J. Math. Ext., 15 (2021).
[11]	H. Orhan, L. I. Cotîrlă, Fekete–Szegö inequalities for some certain subclass of analytic functions defined with Ruscheweyh derivative operator, Axioms, 11 (2022), 560. https://doi.org/10.3390/axioms11100560 doi: 10.3390/axioms11100560
[12]	S. Al-Sa'di, I. Ahmad, S. G. A. Shah, S. Hussain, S. Noor, Fekete–Szegö type functionals associated with certain subclasses of bi-univalent functions, Heliyon, 10 (2024), e28074. https://doi.org/10.1016/j.heliyon.2024.e28074 doi: 10.1016/j.heliyon.2024.e28074
[13]	C. Pommerenke, Univalent functions, Göttingen: Vandenhoeck and Ruprecht, 1975.
[14]	D. Srivastava, S. Porwal, Some sufficient conditions for Poisson distribution series associated with conic regions, Int. J. Adv. Tech. Eng. Sci., 3 (2015), 229–235.
[15]	H. M. Srivastava, G. S. Timilehin, 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. https://doi.org/10.3934/math.2023016 doi: 10.3934/math.2023016
[16]	J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. Math., 17 (1915), 12–22. https://doi.org/10.2307/2007212 doi: 10.2307/2007212
[17]	M. A. Nasr, M. K. Aouf, Radius of convexity for the class of starlike functions of complex order, Bull. Fac. Sci. Assiut Univ. Sect. A, 12 (1983), 153–159.
[18]	M. A. Nasr, M. K. Aouf, Bounded starlike functions of complex order, P. Indian AS-Math. Sci., 92 (1983), 97–102. https://doi.org/10.1007/BF02863012 doi: 10.1007/BF02863012
[19]	M. A. Nasr, M. K. Aouf, Starlike functions of complex order, J. Natural Sci. Math., 25 (1985), 1–12.
[20]	S. S. Miller, P. T. Mocanu, Differential subordinations: Theory and applications, New York, Basel: Marcel Dekker, 2000.
[21]	A. Baricz, S. Ponnusamy, Starlikeness and convexity of generalized Bessel functions, Integr. Transf. Spec. F., 21 (2010), 641–651. https://doi.org/10.1080/10652460903516736 doi: 10.1080/10652460903516736
[22]	E. Deniz, Convexity of integral operators involving generalized Bessel functions, Integr. Transf. Spec. F., 24 (2013), 201–216. https://doi.org/10.1080/10652469.2012.685938 doi: 10.1080/10652469.2012.685938
[23]	E. Deniz, H. Orhan, H. M. Srivastava, Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions, Taiwan. J. Math., 15 (2011), 883–917. https://doi.org/10.11650/twjm/1500406240 doi: 10.11650/twjm/1500406240
[24]	S. Andras, A. Baricz, Monotonicity property of generalized and normalized Bessel functions of complex order, Complex Var. Elliptic, 54 (2009), 689–696. https://doi.org/10.1080/17476930902998936 doi: 10.1080/17476930902998936
[25]	A. Baricz, Generalized Bessel functions of the first kind, Berlin: Springer-Verlag, 2010.
[26]	A. Baricz, E. Deniz, M. Caglar, H. Orhan, Differential subordinations involving generalized Bessel functions, B. Malays. Math. Sci. So., 38 (2015), 1255–1280. https://doi.org/10.1007/s40840-014-0079-8 doi: 10.1007/s40840-014-0079-8
[27]	M. Riyasat, S. Khan, T. Nahid, Quantum algebra $\varepsilon_q(2)$ and 2D $q$-Bessel functions, Rep. Math. Phys., 83 (2019), 191–206. https://doi.org/10.1016/S0034-4877(19)30039-4 doi: 10.1016/S0034-4877(19)30039-4
[28]	M. Riyasat, S. Khan, M. Haneef, Generalized 2D extension of the $q$-Bessel polynomials, Rep. Math. Phys., 90 (2022), 63–80. https://doi.org/10.1016/S0034-4877(22)00051-9 doi: 10.1016/S0034-4877(22)00051-9
[29]	H. M. Srivastava, An introductory overview of Bessel polynomials, the generalized Bessel polynomials and the $q$-Bessel polynomials, Symmetry, 15 (2023), 822. https://doi.org/10.3390/sym15040822 doi: 10.3390/sym15040822
[30]	F. H. Jackson, XI.–On $q$-functions and a certain difference operator, Earth Envi. Sci. T. R. So., 46 (1909), 253–281.
[31]	A. Aral, V. Gupta, R. P. Agarwal, Applications of $q$-calculus in operator theory, New York: Springer, 2013. https://doi.org/10.1007/978-1-4614-6946-9
[32]	M. E. H. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Var. Theor. Appl., 14 (1990), 77–84. https://doi.org/10.1080/17476939008814407 doi: 10.1080/17476939008814407
[33]	S. Mandal, M. M. Soren, Fourth-order differential subordination results for analytic functions involving generalized Bessel functions, TWMS J. Appl. Eng. Math., 15 (2025), 1245–1258.
[34]	A. Pfluger, The Fekete–Szegö inequality by a variational method, Ann. Fenn. Math., 10 (1985), 447–454.
[35]	H. Orhan, E. Deniz, D. Răducanu, The Fekete–Szegö problem for subclasses of analytic functions defined by a differential operator related to conic domains, Comput. Math. Appl., 59 (2010), 283–295. https://doi.org/10.1016/j.camwa.2009.07.049 doi: 10.1016/j.camwa.2009.07.049
[36]	E. Deniz, M. Çağlar, H. Orhan, The Fekete–Szegö problem for a class of analytic functions defined by Dziok–Srivastava operator, Kodai Math. J., 35 (2012), 439–462. https://doi.org/10.2996/kmj/1352985448 doi: 10.2996/kmj/1352985448
[37]	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.
[38]	S. Kanas, H. E. Darwish, Fekete–Szegö problem for starlike and convex functions of complex order, Appl. Math. Lett., 23 (2010), 777–782. https://doi.org/10.1016/j.aml.2010.03.008 doi: 10.1016/j.aml.2010.03.008
[39]	T. Al-Hawary, A. Amourah, B. A. Frasin, Fekete–Szegö inequality for bi-univalent functions by means of Horadam polynomials, Bol. Soc. Mat. Mex., 27 (2021), 79. https://doi.org/10.1007/s40590-021-00385-5 doi: 10.1007/s40590-021-00385-5
[40]	C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann., 64 (1907), 95–115. https://doi.org/10.1007/BF01449883 doi: 10.1007/BF01449883
[41]	R. J. Libera, E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in $\mathcal{P}$, P. Am. Math. Soc., 87 (1983), 251–257. https://doi.org/10.1090/S0002-9939-1983-0681830-8 doi: 10.1090/S0002-9939-1983-0681830-8
[42]	W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, In: Proceedings of the Conference on Complex Analysis, International Press Inc., 1992,157–169.
[43]	M. Arif, M. Raza, H. Tang, S. Hussain, H. Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math., 17 (2019), 1615–1630. https://doi.org/10.1515/math-2019-0132 doi: 10.1515/math-2019-0132
[44]	V. Ravichandran, S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris, 353 (2015), 505–510. https://doi.org/10.1016/j.crma.2015.03.003 doi: 10.1016/j.crma.2015.03.003
[45]	S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal., 2014 (2014), 984135. https://doi.org/10.1155/2014/984135 doi: 10.1155/2014/984135
[46]	S. Porwal, M. A. Kumar, Unified study on starlike and convex functions associated with Poisson distribution series, Afr. Mat., 27 (2016), 1021–1027. https://doi.org/10.1007/s13370-016-0398-z doi: 10.1007/s13370-016-0398-z

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)