Fekete-Szegő Problems and Toeplitz Determinants for Meromorphic Functions Involving q-Calculus in a Cardioid Domain

In this paper, we introduce and investigate a new subclass of meromorphic p-valent functions, denoted by 〖MS〗_(p,q)^* [b;C], defined in the punctured unit disk U^*. The definition of this class is characterized by a subordination condition involving the q-derivative operator and a specific subordinating function, C(z)=1+4/3 z+2/3 z^2, which maps the unit disk onto a Cardioid-shaped domain . We first establish a necessary and sufficient condition for functions to belong to this class using the Hadamard product (convolution). Subsequently, we derive sharp bounds for the first two initial Taylor-Laurent coefficients |a_(1-p) | and |a_(2-p) |. In addition, we also deal with the Fekete-Szegő functional |a_(2-p)-μa_(1-p)^2 | for both real and complex parameters μ. The results obtained in this work generalize several existing findings in the literature and highlight the geometric impact of the Cardioid-shaped domain on meromorphic p-valent starlike functions.

Meromorphic p-valent functions, Starlike function, q-derivative operator

1. Agrawal, S. and Sahoo, S. K., “A generalization of starlike functions of order α,” Hokkaido Mathematical Journal, vol. 46, pp. 15–27, 2017. [Google Scholar] [Crossref]

2. Al-Shbeil, Gong, J., Ray, S., Khan, S., Khan, N., and Alaqad, H., “The properties of meromorphic multivalent q-starlike functions in the janowski domain,” Fractal and Fractional, vol. 7, no. 6, 2023, doi: 10.3390/fractalfract7060438. [Google Scholar] [Crossref]

3. Aouf, M. K., Mostafa, A. O., and Zayed, H. M., “Convolution properties for some subclasses of meromorphic functions of complex order,” Abstract and Applied Analysis, pp. 1–6, 2015. [Google Scholar] [Crossref]

4. Duren, P. L., Univalent functions, vol. 259. in Grundlehren der mathematischen wissenschaften, vol. 259. New York: Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5534-5. [Google Scholar] [Crossref]

5. Fekete, M. and Szegö, G., “Eine bemerkung über univalente funktionen,” Journal of the London Mathematical Society, vol. 1, no. 2, pp. 85–89, 1933. [Google Scholar] [Crossref]

6. Jackson, F. H., “On q-definite integrals,” Quarterly Journal of Pure and Applied Mathematics, vol. 41, pp. 193–203, 1910. [Google Scholar] [Crossref]

7. Jackson, F. H., “On q-functions and a certain difference operator,” Transactions of the Royal Society of Edinburgh, vol. 46, no. 2, pp. 253–281, 1909. [Google Scholar] [Crossref]

8. Kant, S., Gupta, N., and Issar, S., “On hadamard and quasi-hadamard properties for certain subclasses of analytic functions,” Surveys in Mathematics and its Applications, vol. 19, pp. 385–399, 2024. [Google Scholar] [Crossref]

9. Keogh, F. R. and Merkes, E. P., “A coefficient inequality for certain classes of analytic functions,” Proceedings of the American Mathematical Society, vol. 20, no. 1, pp. 8–12, 1969. [Google Scholar] [Crossref]

10. Kumar, S. and Çetinkaya, A., “Coefficient inequalities for certain starlike and convex functions,” Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, pp. 156–171, 2022. [Google Scholar] [Crossref]

11. Kumar, V. and Cho, N. E., “Hermitian–toeplitz determinants for functions with bounded turning,” Turkish Journal of Mathematics, vol. 45, no. 6, pp. 2678–2687, 2021, doi: 10.3906/mat-2104-104. [Google Scholar] [Crossref]

12. Ma, W. C. and Minda, D., “A unified treatment of some special classes of univalent functions,” in Conference proceedings and lecture notes in analysis, 1994, pp. 157–169. [Google Scholar] [Crossref]

13. Marimuthu, K., Yalçın, S., Jayaraman, U., Saravanan, G., and Baskaran, S., “A new family of meromorphic multivalent functions defined by bessel functions,” Montes Taurus Journal of Pure and Applied Mathematics, vol. 6, no. 2, pp. 69–77, 2024. [Google Scholar] [Crossref]

14. Meena, K. P., Bapna, I. B., and Sharma, V., “Certain new subclasses of multivalent meromorphic function,” International Journal of Statistics and Applied Mathematics, vol. 9, no. 5, pp. 46–51, 2024, doi: 10.22271/maths.2024.v9.i5a.1808. [Google Scholar] [Crossref]

15. Miller, S. S. and Mocanu, P. T., Differential subordinations: Theory and applications. New York: Marcel Dekker, Inc., 2000. [Google Scholar] [Crossref]

16. Pommerenke, Ch., Univalent functions. Vandenhoeck & Ruprecht, 1975. [Google Scholar] [Crossref]

17. Rogosinski, W., “On the coefficients of subordinate functions,” Proceedings of the London Mathematical Society, vol. 2, 48, no. 1, pp. 48–82, 1943. [Google Scholar] [Crossref]

18. Seoudy, T. M., “Classes of analytic functions associated with certain integral operator,” Electronic Journal of Mathematical Analysis and Applications, vol. 4, no. 2, pp. 254–258, 2016. [Google Scholar] [Crossref]

19. Sharma, K., Jain, N. K., and Ravichandran, V., “Starlike functions associated with a cardioid,” Afrika Matematika, vol. 28, no. 7, pp. 1013–1039, 2017. [Google Scholar] [Crossref]

20. Sokół, J. and Stankiewicz, J., “On some subclass of starlike functions,” Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka i Fizyka, vol. 19, pp. 101–105, 1996. [Google Scholar] [Crossref]

21. Srivastava, H. M., Arif, M., and Raza, M., “Convolution properties of meromorphically harmonic functions defined by a generalized convolution q-derivative operator,” Mathematics, AIMS, vol. 6, no. 6, pp. 5869–5885, 2021, doi: 10.3934/math.2021347. [Google Scholar] [Crossref]

22. Srivastava, H. M., Hossen, H. M., and Aouf, M. K., “A unified presentation of some classes of meromorphically multivalent functions,” Computers & Mathematics with Applications, vol. 38, no. 11, pp. 63–70, 1999, doi: https://doi.org/10.1016/S0898-1221(99)00285-0. [Google Scholar] [Crossref]

23. Srivastava, H. M., Tahir, M., Khan, B., Darus, M., Khan, N., and Ahmad, Q., “Certain subclasses of meromorphically q-starlike functions associated with the q-derivative operators,” Ukrainian Mathematical Journal, vol. 73, pp. 1462–1477, 2022, doi: 10.1007/s11253-022-02005-5. [Google Scholar] [Crossref]

24. Thulasiram, T., Kalaiselvan, S., Breaz, D., Suchithra, K., and Sudharsan, T. V., “Bounds for hermitian toeplitz and hankel determinants for a certain subclass of analytic functions related to the sine function,” Symmetry, vol. 17, no. 3, p. 362, 2025, doi: 10.3390/sym17030362. [Google Scholar] [Crossref]

25. Ur Rehman, M. S. et al., “Coefficient inequalities for multivalent janowski type q-starlike functions involving certain conic domains,” Axioms, vol. 11, no. 10, p. 494, 2022. [Google Scholar] [Crossref]

• The Impact of Ownership Structure on Dividend Payout Policy of Listed Plantation Companies in Sri Lanka
• Urban Sustainability in North-East India: A Study through the lens of NER-SDG index
• Performance Assessment of Predictive Forecasting Techniques for Enhancing Hospital Supply Chain Efficiency in Healthcare Logistics
• The Fractured Self in Julian Barnes' Postmodern Fiction: Identity Crisis and Deflation in Metroland and the Sense of an Ending
• Impact of Flood on the Employment, Labour Productivity and Migration of Agricultural Labour in North Bihar