Document Type : Research Paper

Authors

1 Faculty of Science and Arts, Department of Mathematics, Düzce University, Düzce, Turkey.

2 Faculty of Science and Arts, Department of Mathematics, TekirdaÄŸ NamÄ±k Kemal University, TekirdaÄŸ, Turkey.

Abstract

In this paper, some conditions have been improved so that the function $g(z)$ is defined as $g(z)=1+\sum_{k\ge 2}^{\infty}a_{n+k}z^{n+k}$, which is analytic in unit disk $U$, can be in more specific subclasses of the $S$ class, which is the most fundamental type of univalent function. It is analyzed some characteristics of starlike and convex functions of order $2^{-r}$.

Keywords

###### ##### References
[1] P.L. Duren, Univalent functions, Springer-Verlag, New York, 1983.

[2] A.W. Goodman, Univalent functions- Vols. I and II, Polygonal Pub., Washington, New Jersey, 1983.

[3] J. Li and S. Owa, Sufficient conditions for starlikeness, Indian J. Pure Appl. Math. 33(9) (2002), pp. 1385–1390.
[4] M. Nunokawa, M. AydoÄŸan, K. Kuroki, I. Yildiz and S. Owa, Some properties concerning close-to-convexity of certain analytic functions, J. Inequal. Appl., 245 (2012), pp. 1-10.
[5] M. Nunokawa, S. Owa, S.K. Lee, M. ObradoviÄ‡, M.K. Aouf, H. Saitoh, A. Ikeda and N. Koike, Sufficient conditions for starlikeness, Chin. J. Math., 24 (1996), pp. 265–271.
[6] M. ObradoviÄ‡ and S. Owa, On certain properties for some classes of starlike functions,     J. Math. Anal. Appl., 145 (1990), pp. 357-364.
[7] S. Owa, On a class of starlike functions II, Chin. J. Math., 19(1) (1982), pp. 29–38.
[8] S. Owa, Sufficient Conditions for Close-to-Convexity, Chin. J. Math., 22(3) (1994), pp. 291-294.
[9] I. Yildiz and A. Akyar, An analytical investigatÄ±on on starlikeness and convexity properties for hypergeometric functions, Turk. J. Math., 44 (2020), pp. 1453-1465.