Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated with -Analogue of the Exponential Function with Complex Order

Nizami Mustafa; Veysel Nezir

Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated with -Analogue of the Exponential Function with Complex Order

Abstract

In this study, by making the use q -analogous of the exponential function, a new subclass of univalent functions is defined. The aim of this study is to find an upper bound estimate for the second Hankel determinant of functions belonging to the defined class. Also, in this study we examine the Zalcman conjecture for the defined class. Moreover, some known results depending on parameter q choices are also discussed.

Keywords

References

ALOTAIBI, A., ARIF, M., ALGHAMDI, M. A., HUSSAIN, S. (2020). Starlikness associated with cosine hyperbolic function. Mathematics 8, 1118.
ARIF, M., TANG, H., HUSSAIN, S., KHAN, H. (2019). Hankel determinant of order three for famliar subsets of analytic functions related with sine function. Open Math., 17, 1615-1630.
BANO, K., RAZA, M. (2020). Starlike functions associated with cosine function. Bull. Iran. Math. Soc. 47, 1513-1532.
BIEBERBACH, L. (1916). Über die Koeffizienten derjenigen Potenzreihen welche eine shlichte Abbidung des Einbeiskreises vermitteln. Sitzungsberichte Preuss. Akad. Der Wiss., 138, pp. 940-955.
DE BRANGES, L. (1985). A proof of the Bieberbach conjecture. Acta Math. 1985, 154, pp. 137-152.
DUREN, P. L. (1983). Univalent Functions. In Grundlehren der Mathematischen Wissenschaften; Springer: New York, NY, USA; Berlin/Heidelberg, Germany; Tokyo, Japan, Volume 259.
HAYMAN, W. K. (1968). On second Hankel determinant of mean univalentfuctions. Proc. Lond. Math. Soc., 3, 77-94.
JANTENG, A., HALIM, S. A., DARUS, M. (2006). Coefficient inequality for a function whose derivative has a poisitive real part. J. Inequal Pure Appl. Math., 7, 1-5.

JANTENG, A., HALIM, S. A., DARUS, M. (2007). Hankel determinant for starlike and convex functions. Int. J. Math. Anal., 1, 619-625.
KÖEBE, P. (1909). Über die Uniformisierrung der algebraishen Kurven, durch automorpher Funktionen mit imaginarer Substitutions gruppe. Nachr. Akad. Wiss. Göttingen Math.-Phys., pp. 68-76.
KUMAR, S. S., ARORA, K. (2020). Starlike functions associated with a petal shaped domain. ArXiv 2020, arXiv: 2010.10072.
LIBERA, R. J., ZLOTKIEWICZ, E. J. (1982). Early coefficients of the inverse of a regular convex function. Proc. Am. Soc. 85, 225-230.
MA, W. C., MINDA, D. (1994). A unified treatment of some special classes of univalent functions. In Peoceedings of the Conference on Complex Analysis, Tianjin, China; Li, Z., Ren, F., Yang, L., Zhang, S., Eds.; Conference Proceedings and Lesture Notes in Analysis; International Press: Cambridge, MA, USA, Volume I, pp. 157-169.
MA, W. (1999). Generalized Zalcman conjecture for starlike and typically real functions. J. Math. Anal. Appl. 234, 328-339.
MENDIRATTA, R., NAGPAL, S., RAVICHANDRAN, V. (2015). On a subclass of strongly starlike functions assiociated with exponential function. Bull. Malays. Math. Soc., 8, pp. 365-386.
MILLER, S. S. (1975). Differential inequalities and Caratheodory functions. Bull. Am. Math. Soc. 81, 79-81.
ÇAĞLAR, M., ORHAN, H., SRIVASTAVA H. M. (2023). Coefficient bounds for q-starlike functions associated with q-Bernoulli numbers. J. Appl. Anal. Comput 13 (4), 2354-2364.
MUSTAFA, N., NEZIR, V., KANKILIÇ. A. (2023). Coefficient estimates for certain subclass of analytic and univalent functions associted with sine hyperbolic function. 13th International Istanbul Scientific Research Congress on Life, Engineering and Applied Sciences on April 29-30, 234-241, Istanbul, Turkey.
MUSTAFA, N., NEZIR, V., KANKILIÇ. A. (2023). The Fekete-Szegö problem for certain subclass of analytic and univalent functions associated with sine hyperbolic function. 13th International Istanbul Scientific Research Congress on Life, Engineering and Applied Sciences on April 29-30, 242-249, Istanbul, Turkey.
MUSTAFA, N., NEZIR, V. (2023). Coefficient estimates and Fekete-Szegö problem for certain subclass of analytic and univalent functions associated with sine hyperbolic function. 13th International Istanbul Scientific Research Congress on Life, Engineering and Applied Sciences on May 1-2, 475-481, Istanbul, Turkey.
MUSTAFA, N., DEMIR, H. A. (2023). Coefficient estimates for certain subclass of analytic and univalent functions with associated with sine and cosine functions. 4th International Black Sea Congress on Modern Scientific Research on June 6-8, 2555-2563, Rize Turkey.
MUSTAFA, N., DEMIR, H. A. (2023). Fekete-Szegö problem for certain subclass of analytic and univalent functions associated with sine and cosine functions. 4th International Black Sea Congress on Modern Scientific Research on June 6-8, 2564-2572, Rize Turkey.
MUSTAFA, N., NEZIR, V., KANKILIÇ. A. (2023). Coefficient estimates for certain subclass of analytic and univalent functions associated with sine hyperbolic function with complex order. Journal of Scientific and Engineering Research, 10(6), 18-25.
MUSTAFA, N., NEZIR, V., KANKILIÇ. A. (2023). The Fekete-Szegö problem for certain subclass of analytic and univalent functions associated with sine hyperbolic function with complex order. Eastern Anatolian Journal of Science, 9(1), 1-6.
MUSTAFA, N., DEMIR, H. A. (2023). Coefficient estimates for certain subclass of analytic and univalent functions with associated with sine and cosine functions with complex order. Journal of Scientific and Engineering Research, 10(6), 131-140.
OBRADOVIC, M., TUNESKI, N. (2019). Hankel determinants of second and third order for the class S of univalent functions. arXiv 2019, arXiv:1912.06439.
POMMARENKE, C. (1975). Univalent Function. Vanderhueck and Ruprecht: Göttingen, German. POMMARENKE, C. (1967). On the Hankel determiants of univalent functions. Mathematika, 14, 108-112.
SOKOL, J. (2011). A certain class of starlike functions. Comput. Math. Appl., 62, pp. 611-619.
ULLAH, K., ZAINAB, S., ARIF, M., DARUS, M., SHUTAYI, M. (2021). Radius Problems for Starlike Functions Associated with the Tan Hyperbolic Function. J. Funct. Spaces, Article ID 9967640.

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Nizami Mustafa ^*
0000-0002-2758-0274
Türkiye

Veysel Nezir
0000-0001-9640-8526
Türkiye

Publication Date

October 18, 2026

Submission Date

July 14, 2026

Acceptance Date

August 9, 2026

Published in Issue

Year 2026 Volume: 12 Number: 1

IZ

https://izlik.org/JA66PE53YP

Cite

RIS / Bibtex

APA

Mustafa, N., & Nezir, V. (2026). Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated with -Analogue of the Exponential Function with Complex Order. Eastern Anatolian Journal of Science, 12(1), 25-31. https://izlik.org/JA66PE53YP

AMA

1.Mustafa N, Nezir V. Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated with -Analogue of the Exponential Function with Complex Order. Eastern Anatolian Journal of Science. 2026;12(1):25-31. https://izlik.org/JA66PE53YP

Chicago

Mustafa, Nizami, and Veysel Nezir. 2026. “Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated With -Analogue of the Exponential Function With Complex Order”. Eastern Anatolian Journal of Science 12 (1): 25-31. https://izlik.org/JA66PE53YP.

EndNote

Mustafa N, Nezir V (October 1, 2026) Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated with -Analogue of the Exponential Function with Complex Order. Eastern Anatolian Journal of Science 12 1 25–31.

IEEE

[1]N. Mustafa and V. Nezir, “Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated with -Analogue of the Exponential Function with Complex Order”, Eastern Anatolian Journal of Science, vol. 12, no. 1, pp. 25–31, Oct. 2026, [Online]. Available: https://izlik.org/JA66PE53YP

ISNAD

Mustafa, Nizami - Nezir, Veysel. “Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated With -Analogue of the Exponential Function With Complex Order”. Eastern Anatolian Journal of Science 12/1 (October 1, 2026): 25-31. https://izlik.org/JA66PE53YP.

JAMA

1.Mustafa N, Nezir V. Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated with -Analogue of the Exponential Function with Complex Order. Eastern Anatolian Journal of Science. 2026;12:25–31.

MLA

Mustafa, Nizami, and Veysel Nezir. “Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated With -Analogue of the Exponential Function With Complex Order”. Eastern Anatolian Journal of Science, vol. 12, no. 1, Oct. 2026, pp. 25-31, https://izlik.org/JA66PE53YP.

Vancouver

1.Nizami Mustafa, Veysel Nezir. Upper Bound Estimate of the Second Hankel Determinant for Certain Subclass of Starlike Functions Associated with -Analogue of the Exponential Function with Complex Order. Eastern Anatolian Journal of Science [Internet]. 2026 Oct. 1;12(1):25-31. Available from: https://izlik.org/JA66PE53YP