Fekete–SzegÖ inequalities and initial coefficient bounds for Bi-univalent functions involving fractional q-differential operator subordination to q-hermite polynomials
Abstract views: 0
/ 
PDF downloads: 0
Keywords:
q-Hermite Polynomials; Fekete–Szegö; Starlike; Fractional q-calculus; Convex functions.Abstract
In this paper, we introduce new subclasses of bi-univalent functions in the open unit disk defined via the q-fractional difference operator related to q-Hermite polynomials. Using subordination, we define the starlike class and the convex class . For functions in these classes, we derive upper bounds for the first two Taylor coefficients |a2| and |a3|, and we establish upper Fekete–Szegö inequalities for 2 aa 32 - b and 2 aa 32 - m , extending classical results to the q-fractional setting. We focus particularly on comparing the initial coefficients of these functions, providing insight into their geometric behavior and the influence of the q-fractional operator on the coefficient bounds.
References
Tayyah, A.S.; Atshan, W.G. Differential Subordination and Superordination Results for p-Valent Analytic Functions Associated with (r, k)-Srivastava Fractional Integral Calculus. MethodsX 2024, 13, 103079. https://doi.org/10.1016/j.mex.2024.103079
Hristov, J. Approximate Solutions to Fractional Subdiffusion Equations. Eur. Phys. J. Spec. Top. 2011, 193, 229–243.
Al-Omari, S.Q. On q-Analogues of the Natural Transform of Certain q-Bessel Functions and Some Application. Filomat 2017, 31, 2587–2598.
Lupas, A. A q-Analogue Bernstein Operator. In Seminar on Numerical and Statistical Calculus; University of Cluj-Napoca: Cluj-Napoca, Romania, 1987; Volume 9, pp. 85–92.
Jackson, F.H. On q-Definite Integrals on q-Functions and a Certain Difference Operator. Trans. R. Soc. Edinb. 1908, 46, 253–281.
El-Ityan, M.; Al-Hawary, T.; Frasin, B.A.; Aldawish, I. A New Subclass of BiUnivalent Functions Defined by Subordination to Laguerre Polynomials and the (p, q)-Derivative Operator. Symmetry 2025, 17, 982.
El-Ityan, M.; Al-Hawary, T.; Frasin, B.A.; Aldawish, I. Third Hankel Determinant for a Subclass of Bi-Univalent Functions Related to Balloon-Shaped Domain. AIMS Mathematics 2025, 10(11), 25452–25468. doi: 10.3934/math.20251127.
Tayyah, A.S.; Atshan, W.G.; Yalçın, S. Third-Order Differential Subordination and Superordination Results for p-Valent Analytic Function Involving Fractional Derivative Operator. Mathematical Methods in the Applied Sciences 2025.
Lewin, M. On a coefficient problem for bi-univalent functions. Proc. Amer. Math. Soc. 1967, 18, 63–68.
Brannan, D.A.; Clunie, J.G. (Editors). Aspects of Contemporary Complex Analysis; Academic Press: London, 1980.
Brannan, D.A.; Clunie, J.; Kirwan, W.E. Coefficient estimates for a class of star-like functions. Canad. J. Math. 1970, 22, 476-485.
Brannan, D.A.; Taha, T.S. On some classes of bi-univalent functions. Studia Univ. Babeş-Bolyai Math. 1986, 31(2), 70–77.
Netanyahu, E. The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1. Arch. Rational Mech. Anal. 1969, 32, 100–112.
Taha, T.S. Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981.
Lupaş, A.A.; Tayyah, A.S.; Sokół, J. Sharp bounds on Hankel determinants for starlike functions defined by symmetry with respect to symmetric domains. Symmetry 2025, 17, 1244.
Ma, W.; Minda, D. A unified treatment of some special cases of univalent functions. In Proc. Conf. on Complex Analysis, Nankai Institute of Mathematics, Tianjin, China; International Press: Cambridge, MA, USA, 1994; Conf. Proc. Lect. Notes Anal. 1, pp. 157–169.
Sokol, J. Radius problems in the class YL. Appl. Math. Comput. 2009, 214 (2), 569– 573.
Raza, M.; Malik, S.N. Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli. J. Inequal. Appl. 2013, Article ID 412.
Raina, R.K.; Sokol, J. On coefficient estimates for a class of starlike functions. Hacet. J. Math. Stat. 2015, 44 (6), 1427–1433.
Tayyah, A.S.; Atshan, W.G. Starlikeness and Bi-Starlikeness Associated with a New Carath ́eodory Function. Journal of Mathematical Sciences 2025, 290, 232–256. https://doi.org/10.1007/s10958-025-07604-8
Arif, M.; Raza, M.; Tang, H.; Hussain, S.; Khan, H. Hankel determinant of order three for familiar subsets of analytic functions related with sine function. Open Math. 2019, 17, 1615–1630.
Jackson, F.H. On q-Definite Integrals. Quart. J. Pure Appl. Math. 1910, 41, 193–203.
Khan, B.; Liu, Z.G.; Srivastava, H.M.; Khan, N.; Darus, M.; Tahir, M. A Study of Some Families of Multivalent q-Starlike Functions Involving Higher-Order q- Derivatives. Mathematics 2020, 8, 1490.
Wanas, A.K.; El-Ityan, M.; Tayyah, A.S.; Catas, A. Certain Subclasses of BiUnivalent Functions Involving Caputo Fractional Derivatives with Bounded Boundary Rotation. Mathematics 2025, 13, 3563.
Sharma, K.; Jain, N.K.; Ravichandran, V. Starlike functions associated with a cardioid. Afr. Mat. 2016, 27 (5–6), 923–939.
Amini, E.; Al-Omari, S.; Nonlaopon, K.; Baleanu, D. Estimates for Coefficients of BiUnivalent Functions Associated with a Fractional q-Difference Operator. Symmetry 2022, 14, 879. https://doi.org/10.3390/sym14050879.
Osburn, R.; Schneider, C. Gaussian Hypergeometric Series and Supercongruences. Math. Comput. 2009, 78, 275–292.
Khan, B.; Liu, Z.G.; Srivastava, H.M.; Khan, N.; Tahir, M. Applications of Higher-Order Derivatives to Subclasses of Multivalent q-Starlike Functions. Maejo Int. J. Sci. Technol. 2021, 15, 61–72.
Srivastava, H.M.; Mishra, A.K.; Gochhayat, P. Certain Subclasses of Analytic and Bi-Univalent Functions. Appl. Math. Lett. 2010, 23, 1188–1192.
Babalola, K.O. New Subclasses of Analytic and Univalent Functions Involving Certain Convolution Operator. Math. Tomé 2008, 50, 3–12.
Ismail, E.H.; Stanton, D.; Viennot, G. The Combinatorics of q-Hermite Polynomials and the Askey–Wilson Integral. Eur. J.Combin. 1987, 8, 379–392.
Ruscheweyh, S.T. New Criteria for Univalent Functions. Proc. Am. Math. Soc. 1975, 49, 109–115.
Rao, P.; Vyas, M.; Chavda, N.D. Eigenstate Structure in Many-Body Bosonic System: Analysis Using Random Matrices and q-Hermite Polynomial. arXiv 1933, arXiv:2111.08820v1.
Chavda, N.D. Average-Fluctuation Separation in Energy Levels in Quantum ManyParticle Systems with k-Body Interactions Using q-Hermite Polynomials. arXiv 2021, arXiv:2111.12087v1.
Alsoboh, A, Amourah, A, Al-Maqbali, A, Alnajar, O, Awad, F, Sasa, T, A Class of Bi-Univalent Functions Associated with Shell-Like Geometries and the q-Fibonacci Analogue, European Journal of Pure and Applied Mathematics 2025, 18 (4), 6689–6689
Alsoboh, A, Amourah, A, Almashrafi, K, A Study on Bi-Univalent Functions of Complex Order Arising from the q-Fibonacci Analogue, European Journal of Pure and Applied Mathematics 2025, 18 (4), 7065–7065
Alsoboh, A, Almalkawi, A, Amourah, A, Al Mashrafi, K, Sasa, T, Coefficient Problems for Bi-Univalent Functions via q-Rabotnov Kernels and q–Fibonacci Subordination, European Journal of Pure and Applied Mathematics 2025, 18 (4), 7125–7125
Al-Shbeil, I.; Cătaş, A.; Srivastava, H.M.; Aloraini, N. Coefficient Estimates of New Families of Analytic Functions Associated with q -Hermite Polynomials. Axioms 2023, 12, 52.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Results in Nonlinear Analysis
This work is licensed under a Creative Commons Attribution 4.0 International License.
