Bounds for Toeplitz Determinants and Related Inequalities for a New Subclass of Analytic Functions
Abstract
:1. Introduction
2. Preliminaries
- If and , then , which is the familiar subclass of starlike functions.
- If and , then , the family of normalized univalent convex functions.
3. Main Results
4. Toeplitz Determinant
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Kanas, S.; Wisniowska, A. Conic regions and k-uniform convexity. J. Comput. Appl. Math. 1999, 105, 327–336. [Google Scholar] [CrossRef]
- Kanas, S.; Wisniowska, A. Conic domains and starlike functions. Rev. Roum. Math. Pures Appl. 2000, 45, 647–657. [Google Scholar]
- Goodman, A.W. On uniformly convex functions. Ann. Pol. Math. 1991, 56, 87–92. [Google Scholar] [CrossRef]
- Wang, Z.G.; Jiang, Y.P. On certain subclasses of close to-convex and quasi-convex functions with respect to 2k-symmetric conjugate points. J. Math. Appl. 2007, 29, 167–179. [Google Scholar]
- Alexander, J.W. Functions which map the interior of the unit circle upon simple regions. Ann. Math. 1915, 17, 12–22. [Google Scholar] [CrossRef]
- Jackson, F.H. On q-functions and a certain difference operator. Earth Environ. Sci. Trans. R. Soc. Edinb. 1909, 46, 253–281. [Google Scholar] [CrossRef]
- Jackson, F.H. On q-definite integrals. Quart. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
- Ismail, M.E.H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var. Theory Appl. 1990, 14, 77–84. [Google Scholar] [CrossRef]
- Kanas, S.; Răducanu, D. Some class of analytic functions related to conic domains. Math. Slovaca 2014, 64, 1183–1196. [Google Scholar] [CrossRef]
- Noor, K.I.; Riaz, S.; Noor, M.A. On q-Bernardi integral operator. TWMS J. Pure Appl. Math. 2017, 8, 3–11. [Google Scholar]
- Sălăgean, G.S. Subclasses of univalent functions. In Lecture Notes in Mathematics; Springer: Berlin, Germany, 1983. [Google Scholar]
- Govindaraj, M.; Sivasubramanian, S. On a class of analytic functions related to conic domains involving q-calculus. Anal. Math. 2017, 43, 457–487. [Google Scholar] [CrossRef]
- Cotîrlǎ, L.I.; Murugusundaramoorthy, G. Starlike functions based on Ruscheweyh q-differential operator defined in Janowski domain. Fractal Fract. 2023, 7, 148. [Google Scholar] [CrossRef]
- Aouf, M.K.; Murugusundaramoorthy, G. On a subclass of uniformly convex functions defined by the Dziok-Srivastava operator. Austral. J. Math. Anal. Appl. 2008, 5, 3. [Google Scholar]
- Tang, H.; Vijaya, K.; Murugusundaramoorthy, G.; Sivasubramanian, S. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator. Open Math. 2021, 19, 329–337. [Google Scholar] [CrossRef]
- Srivastava, H.M. Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A Sci. 2020, 44, 327–344. [Google Scholar] [CrossRef]
- Gasper, G.; Rahman, M. Basic hypergeometric Series Encyclopedia of Mathematics and Its Applications; Cambridge University Press: Cambridge, UK, 1990; Volume 35. [Google Scholar]
- Ruscheweyh, S. New criteria for univalent functions. Proc. Am. Math. Soc. 1975, 49, 109–115. [Google Scholar] [CrossRef]
- Noonan, J.W.; Thomas, D.K. On the second Hankel derminant of areally mean p-valent functions. Trans. Am. Math. Soc. 1976, 223, 337–346. [Google Scholar]
- Dienes, P. The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable; New York-Dover Publishing Company: Mineola, NY, USA, 1957. [Google Scholar]
- Cantor, D.G. Power series with integral coefficients. Bull. Am. Math. Soc. 1963, 69, 362–366. [Google Scholar] [CrossRef]
- Edrei, A. Sur les determinants recurrents et less singularities d’une fonction donée por son developpement de Taylor. Comput. Math. 1940, 7, 20–88. [Google Scholar]
- Polya, G.; Schoenberg, I.J. Remarks on de la Vallee Poussin means and convex conformal maps of the circle. Pac. J. Math. 1958, 8, 259–334. [Google Scholar] [CrossRef]
- Janteng, A.; Abdul-Halim, S.; Darus, M. Coefficient inequality for a function whose derivative has positive real part. J. Inequal. Pure Appl. Math. 2006, 7, 50. [Google Scholar]
- Mahmood, S.; Srivastava, H.M.; Khan, N.; Ahmad, Q.Z.; Khan, B.; Ali, I. Upper bound of the third Hankel determinant for a subclass of q-starlike functions. Symmetry 2019, 11, 347. [Google Scholar] [CrossRef]
- Mishra, A.K.; Gochhayat, P. Second, Hankel determinant for a class of analytic functions defined by fractional derivative. Internat. J. Math. Math. Sci. 2008, 2008, 153280. [Google Scholar] [CrossRef]
- Raza, M.; Malik, S.N. Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli. J. Inequal. Appl. 2013, 2013, 412. [Google Scholar] [CrossRef]
- Singh, G.; Singh, G. On the second Hankel determinant for a new subclass of analytic functions. J. Math. Sci. Appl. 2014, 2, 1–3. [Google Scholar]
- Ayinla, R.; Bello, R. Toeplitz determinants for a subclass of analytic functions. J. Progress. Res. Math. 2021, 18, 99–106. [Google Scholar]
- Radhika, V.; Sivasubramanian, S.; Murugusundaramoorthy, G.; Jahangiri, J.M. Toeplitz matrices whose elements are the coefficients of functions with bounded boundary rotation. J. Complex Anal. 2016, 2016, 4960704. [Google Scholar] [CrossRef]
- Radhika, V.; Sivasubramanian, S.; Murugusundaramoorthy, G.; Jahangiri, J.M. Toeplitz matrices whose elements are coefficients of Bazilevič functions. Open Math. 2018, 16, 1161–1169. [Google Scholar] [CrossRef]
- Ramachandran, C.; Kavitha, D. Toeplitz determinant for some subclasses of analytic functions. Glob. J. Pure Appl. Math. 2017, 13, 785–793. [Google Scholar]
- Srivastava, H.M.; Ahmad, Q.Z.; Khan, N.; Khan, N.; Khan, B. Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain. Mathematics 2019, 7, 181. [Google Scholar] [CrossRef]
- Tang, H.; Khan, S.; Hussain, S.; Khan, N. Hankel and Toeplitz determinant for a subclass of multivalent q-starlike functions of order α. AIMS Math. 2021, 6, 5421–5439. [Google Scholar] [CrossRef]
- Wanas, A.K.; Sakar, F.M.; Oros, G.I.; Cotîrlă, L.-I. Toeplitz Determinants for a Certain Family of Analytic Functions Endowed with Borel Distribution. Symmetry 2023, 15, 262. [Google Scholar] [CrossRef]
- Ma, W.; Minda, D. A unified treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis; Li, Z., Ren, F., Yang, L., Zhang, S., Eds.; International Press: New York, NY, USA, 1994; pp. 157–169. [Google Scholar]
- Duren, P.L. Univalent Functions; Grundlehren der Mathematischen Wissenschaften; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1983; Volume 259. [Google Scholar]
- Hayami, T.; Owa, S. Generalized Hankel determinant for certain classes. Int. J. Math. Anal. 2010, 4, 2573–2585. [Google Scholar]
- Ali, M.F.; Thomas, D.K.; Vasudevarao, A. Toeplitz determinants whose elements are the coefficients of analytic and univalent functions. Bull. Austral. Math. Soc. 2018, 97, 253–264. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Tang, H.; Gul, I.; Hussain, S.; Noor, S. Bounds for Toeplitz Determinants and Related Inequalities for a New Subclass of Analytic Functions. Mathematics 2023, 11, 3966. https://doi.org/10.3390/math11183966
Tang H, Gul I, Hussain S, Noor S. Bounds for Toeplitz Determinants and Related Inequalities for a New Subclass of Analytic Functions. Mathematics. 2023; 11(18):3966. https://doi.org/10.3390/math11183966
Chicago/Turabian StyleTang, Huo, Ihtesham Gul, Saqib Hussain, and Saima Noor. 2023. "Bounds for Toeplitz Determinants and Related Inequalities for a New Subclass of Analytic Functions" Mathematics 11, no. 18: 3966. https://doi.org/10.3390/math11183966
APA StyleTang, H., Gul, I., Hussain, S., & Noor, S. (2023). Bounds for Toeplitz Determinants and Related Inequalities for a New Subclass of Analytic Functions. Mathematics, 11(18), 3966. https://doi.org/10.3390/math11183966