On the Pseudo Starlike and Pseudo Convex Bi-univalent Function Classes of Complex Order

Abstract

In this paper, we defined a new subclass of starlike and convex bi-univalent functions and examine some geometric properties this function class. For this defination class, we gave some coefficient estimates and solve Fekete-Sezöge problem.

Keywords

• Starlike function
• convex function
• pseudo starlike function
• pseudo convex function

References

1. ALOTAIBI, A., ARIF, M., ALGHAMDI, M. A., HUSSAIN, S. (2020) Starlikness associated with cosine hyperbolic function. Mathematics. 8(7), p. 1118.
2. ARIF, M., AHMAD, K., LIU, J.-L., SOKOL, J. (2019) A new class of analytic functions associated with Salagean operator. Hindawi Journal of Function Spaces. Article ID 6157394, 8 pages https://doi.org/10.1155/2019/6157394. BANO, K., RAZA, M. (2020) Starlike functions associated with cosine function. Bull. Iran. Math. Soc. 47, pp. 1513-1532.
3. BRANNAN, D. A., KIRWAN, W. E. (1969) On some classes of bounded univalent functions. J. Lond. Math. Soc. 2, pp. 431-443.
4. BIEBERBACH, L. (1916) Über die Koeffizienten derjenigen Potenzreihen welche eine schlichte Abbidung des Einbeiskreises vermitteln. Sitzungsberichte Preuss. Akad. Der Wiss. 138, pp. 940- 955.
5. CARATHEODORY, C. (1907) Über den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene werte nicht annehmen. Math. Ann. 64, pp.95-115.
6. CHO, N. E., KUMAR, V., KUMAR, S. S., RAVICHANDRAN, V. (2019) Radius problems for starlike functions associated with the sine function. Bull. Iran. Math. Soc. 45, pp. 213-232.
7. DE BRANGES, L. (1985) A proof of the Bieberbach conjecture. Acta Math. 154, pp. 137-152.
8. DUREN, P. L. (1983) Univalent Functions. In Grundlehren der Mathematischen Wissenschaften, New York, Berlin, Heidelberg and Tokyo, Springer- Verlag, Volume 259.

9. FRASIN, B. AND AOUF, M.K. (2011) New Subclasses of Bi-univalent Functions. Applied Mathematics Letters. 24(9), pp. 1569-1573.
10. JANOWSKI, W. (1970) Extremal problems for a family of functions with positive real part and for some related families. Ann. Pol. Math. 23, pp. 159-177.
11. SOKOL, J. A certain class of starlike functions. Comput. Math. Appl. 2011, 62, pp. 611-619.
12. KANKILIÇ. A., MUSTAFA, N. (2024a) On the Pseudo Starlike Bi-univalent Function Class of Complex Order. Journal of Scientific and Engineering Research. 11(5), pp. 26-34.
13. KANKILIÇ. A., MUSTAFA, N. (2024b) On the Pseudo Convex Bi-univalent Function Class of Complex Order. Journal of Scientific and Engineering Research. 11(7). pp. 259-267.
14. 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.
15. KUMAR, S. S., ARORA, K. (2020) Starlike functions associated with a petal shaped domain. arXiv: 2010.10072.
16. MENDIRATTA, R., NAGPAL, S., RAVICHANDRAN, V. (2015) On a subclass of strongly starlike functions associated with exponential function. Bull. Malays. Math. Soc. 38, pp. 365-386.
17. MILLER, S. S. (1975) Differential inequalities and Caratheodory functions. Bull. Am. Math. Soc. 81, pp. 79-81.
18. MUSTAFA, N., NEZIR, V., KANKILIÇ. A. (2023a) Coefficient estimates 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, pp.234-241, Istanbul, Turkey.
19. MUSTAFA, N., NEZIR, V., KANKILIÇ. A. (2023b) 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, pp. 242-249, Istanbul, Turkey.
20. 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, pp. 475-481, Istanbul, Türkiye.
21. MUSTAFA, N., DEMIR, H. A. (2023a) 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, pp. 2555- 2563, Rize Turkey.
22. MUSTAFA, N., DEMIR, H. A. (2023b) 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, pp. 2564-2572, Rize Turkey.
23. MUSTAFA, N., NEZIR, V., KANKILIÇ. A. (2023c) 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), pp.18-25.
24. MUSTAFA, N., NEZIR, V., KANKILIÇ. A. (2023d) 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), pp. 1-6.
25. 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), pp.131-140.
26. SHARMA, K., JAIN, N. K., RAVICHANDRAN, V. (2016) Starlike function associated with a cardioid. Afr. Math. 27, pp. 923-939.
27. SHI, L., SRIVASTAVA, H. M. ARIF, M., HUSSAIN, S., KHAN, H. (2019) An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function. Symmetry. 11, p.598.
28. SOKOL, J., STANKIEWCZ, J. (1996) Radius of convexity of some subclasses of strongly starlike functions. Zesz. Nauk. Politech. Rzesz. Math. vol.19, pp.101–105.
29. ULLAH, K., ZAINAB, S., ARIF, M., DARUS, M., SHUTAYI, M. (2021) Radius Problems for Starlike Functions Associated with the Tan Hyperbolic Function. Hindawi Journal of Function Spaces.2021, 9967640.