Results on Toeplitz Determinants for Subclasses of Analytic Functions Associated to q-Derivative Operator
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Keywords:
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Analytic Functions, Toeplitz Determinant, Quantum (or q-) Calculus, q-Derivative Operator
Abstract
An analytic function, also known as a holomorphic function, is a complex-valued function that is differentiable at every point within a given domain. In other words, a function f (z) is analytic in a domain U if it has a derivative f′(z) at every point z in U. Let A represent the set of functions f that are analytic within the open unit disk D = {z ∈ ℂ : |z| < 1}. These functions possess a normalized Taylor-Maclaurin series expansion written in the form f (z) = z + Í∞ n=2 an z n where an ∈ ℂ, n = 2, 3, . . .. In recent years, the field of q-calculus has gained significant attention and research interest among mathematicians. The applications of this field are broadly applied in numerous subdivisions of physics and mathematics. In this research, we assume that S∗q and ℝq are subclasses of analytic functions obtained by applying the q-derivative operator. The objective of this paper is to obtain estimates for coefficient inequalities and Toeplitz determinants whose elements are the coefficients an for f ∈ S∗q and f ∈ Rq .
References
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Buyankara, M. and M. Çağlar (2023). Hankel and Toeplitz Determinants for a Subclass of Analytic Functions. Matematychni Studii, 60(2); 132–137
Choo, C. P. and A. Janteng (2013). Estimate on the Second Hankel Functional for a Subclass of Close-To-Convex Functions with Respect to Symmetric Points. International Journal of Mathematics Analysis, 7; 781–788
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Hern, A. L. P., A. Janteng, and R. Omar (2020). Hankel Determinant H2(3) for Certain Subclasses of Univalent Functions. Mathematics and Statistics, 8(5); 566–569
Huey, K. S., A. Janteng, J. Janteng, and A. L. P. Hern (2023). Second Hankel Determinant of Bi univalent Functions. Malaysian Journal of Fundamental and Applied Sciences, 19(2); 269–279
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Karahuseyin, Z., S. Altinkaya, and S. Yalçin (2017). On H3 (1) Hankel Determinant for Univalent Functions Defined by Using q-Derivative Operator. TJMM, 9; 25–33
Radhika, V., J. M. Jahangiri, S. Sivasubramanian, and G. Murugusundaramoorthy (2018). Toeplitz Matrices Whose Elements Are Coefficients of Bazilevič Functions. Open Mathematics, 16(1); 1161–1169
Ramachandran, C. and D. Kavitha (2017). Toeplitz Determinant for Some Subclasses of Analytic Functions. Global Journal of Pure and Applied Mathematics, 13(2); 785–793
Rasheed, A., Bello, and Risikat (2023). Toeplitz Determinants for a Subclass of Analytic Functions. 18; 99–106
Sivasubramanian, S., M. Govindaraj, and G. Murugusundaramoorthy (2016). Toeplitz Matrices Whose Elements Are the Coefficients of Analytic Functions Belonging to Certain Conic Domains. International Journal of Pure and Applied Mathematics, 109(10); 39–49
Soh, S. C., D. Mohamad, and H. Dzubaidi (2021). Coefficient Estimates of Toeplitz Determinant for a Certain Class of Close-to-Convex Functions. Malaysian Journal of Fundamental and Applied Sciences, 17; 670–677
Srivastava, H. M., Q. Z. Ahmad, N. Khan, N. Khan, and B. Khan (2019). Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain. Mathematics, 7(2); 181
Sun, Y., M. Arif, K. Ullah, L. Shi, and M. I. Faisal (2023). Hankel Determinant for Certain New Classes of Analytic Functions Associated the Activation Functions. Heliyon, 9(11); 1–12
Tang, H., I. Gul, S. Hussain, and S. Noor (2023). Bounds for Toeplitz Determinants and Related Inequalities for a New Subclass of Analytic Functions. Mathematics, 11(18); 3966
Tang, H., S. Khan, S. Hussain, and N. Khan (2021). Hankel and Toeplitz Determinant for a Subclass of Multivalent qStarlike Functions of Order. AIMS Math, 6; 5421–5439
Wahid, N. H. A. A., D. Mohamad, N. M. Kamarozzaman, and A. A. Shahminan (2022). Toeplitz Determinants for the Class of Functions with Bounded Turning. European Journal of Pure and Applied Mathematics, 15(4); 1937–1947
Wanas, A. K., F. M. Sakar, G. I. Oros, and L.-I. Cotîrlă (2023). Toeplitz Determinants for a Certain Family of Analytic Functions Endowed with Borel Distribution. Symmetry, 15(2); 262
Al-shbeil, I., J. Gong, S. Khan, N. Khan, A. Khan, M. F. Khan, and A. Goswami (2022). Hankel and Symmetric Toeplitz Determinants for a New Subclass of q-Starlike Functions. Fractal and Fractional, 6(11); 658
Ali, M. F., D. Thomas, and A. Vasudevarao (2018). Toeplitz Determinants Whose Elements Are the Coefficients of Analytic and Univalent Functions. Bulletin of the Australian Mathematical Society, 97(2); 253–264
Ayinla, R. and R. Bello (2021). Toeplitz Determinants for a Subclass of Analytic Functions. Journal of Progressive Research in Mathematics, 18; 99–106
Bansal, D. (2013). Upper Bound of Second Hankel Determinant for a New Class of Analytic Functions. Applied Mathematics Letters, 26(1); 103–107
Buyankara, M. and M. Çağlar (2023). Hankel and Toeplitz Determinants for a Subclass of Analytic Functions. Matematychni Studii, 60(2); 132–137
Choo, C. P. and A. Janteng (2013). Estimate on the Second Hankel Functional for a Subclass of Close-To-Convex Functions with Respect to Symmetric Points. International Journal of Mathematics Analysis, 7; 781–788
Duren, P. L. (1983). Univalent Functions. Springer New York, NY Efraimidis, I. (2016). A Generalization of Livingston’s Coefficient Inequalities for Functions with Positive Real Part. Journal of Mathematical Analysis and Applications, 435(1); 369–379
Hern, A. L. P., A. Janteng, and R. Omar (2020). Hankel Determinant H2(3) for Certain Subclasses of Univalent Functions. Mathematics and Statistics, 8(5); 566–569
Huey, K. S., A. Janteng, J. Janteng, and A. L. P. Hern (2023). Second Hankel Determinant of Bi univalent Functions. Malaysian Journal of Fundamental and Applied Sciences, 19(2); 269–279
Jackson, F. (1910). On a q-Definite, Integrals. Quarterly Journal of Pure and Applied Mathematics, 41; 193–203
Jackson, F. H. (1909). On q-Functions and a Certain Difference Operator. Earth and Environmental Science Transactions of the Royal Society of Edinburgh, 46(2); 253–281
Janteng, A., S. A. Halim, and M. Darus (2007). Hankel Determinant for Starlike and Convex Functions. International Journal of Mathematical Analysis, 1(13); 619–625
Karahuseyin, Z., S. Altinkaya, and S. Yalçin (2017). On H3 (1) Hankel Determinant for Univalent Functions Defined by Using q-Derivative Operator. TJMM, 9; 25–33
Radhika, V., J. M. Jahangiri, S. Sivasubramanian, and G. Murugusundaramoorthy (2018). Toeplitz Matrices Whose Elements Are Coefficients of Bazilevič Functions. Open Mathematics, 16(1); 1161–1169
Ramachandran, C. and D. Kavitha (2017). Toeplitz Determinant for Some Subclasses of Analytic Functions. Global Journal of Pure and Applied Mathematics, 13(2); 785–793
Rasheed, A., Bello, and Risikat (2023). Toeplitz Determinants for a Subclass of Analytic Functions. 18; 99–106
Sivasubramanian, S., M. Govindaraj, and G. Murugusundaramoorthy (2016). Toeplitz Matrices Whose Elements Are the Coefficients of Analytic Functions Belonging to Certain Conic Domains. International Journal of Pure and Applied Mathematics, 109(10); 39–49
Soh, S. C., D. Mohamad, and H. Dzubaidi (2021). Coefficient Estimates of Toeplitz Determinant for a Certain Class of Close-to-Convex Functions. Malaysian Journal of Fundamental and Applied Sciences, 17; 670–677
Srivastava, H. M., Q. Z. Ahmad, N. Khan, N. Khan, and B. Khan (2019). Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain. Mathematics, 7(2); 181
Sun, Y., M. Arif, K. Ullah, L. Shi, and M. I. Faisal (2023). Hankel Determinant for Certain New Classes of Analytic Functions Associated the Activation Functions. Heliyon, 9(11); 1–12
Tang, H., I. Gul, S. Hussain, and S. Noor (2023). Bounds for Toeplitz Determinants and Related Inequalities for a New Subclass of Analytic Functions. Mathematics, 11(18); 3966
Tang, H., S. Khan, S. Hussain, and N. Khan (2021). Hankel and Toeplitz Determinant for a Subclass of Multivalent qStarlike Functions of Order. AIMS Math, 6; 5421–5439
Wahid, N. H. A. A., D. Mohamad, N. M. Kamarozzaman, and A. A. Shahminan (2022). Toeplitz Determinants for the Class of Functions with Bounded Turning. European Journal of Pure and Applied Mathematics, 15(4); 1937–1947
Wanas, A. K., F. M. Sakar, G. I. Oros, and L.-I. Cotîrlă (2023). Toeplitz Determinants for a Certain Family of Analytic Functions Endowed with Borel Distribution. Symmetry, 15(2); 262
Authors
Nurali, N. B., & Janteng, A. (2024). Results on Toeplitz Determinants for Subclasses of Analytic Functions Associated to q-Derivative Operator. Science and Technology Indonesia, 9(2), 354–358. https://doi.org/10.26554/sti.2024.9.2.354-358
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