# Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function

• Hasan Şahin Department of Mathematics, Faculty of Arts and Sciences, Duzce University, Turkey
• İsmet Yildiz Department of Mathematics, Faculty of Arts and Sciences, Duzce University, Turkey
Keywords: Analytic function, convexity, unit disk

### Abstract

UDC 517.5

Investigation of the theory of complex functions  is one of the most fascinating aspects of theory of complex analytic functions of one variable.  It has a huge impact on all areas of mathematics.  Many mathematical concepts are explained when viewed through the theory  of complex functions. Let $f(z)\in A,$ $f(z)=z+\sum_{n\geq 2}^{\infty }a_{n}z^{n} ,$  be an analytic function in the open unit disc  $U=\left\{z\colon |z|<1,\ z\in \mathbb{C}\right\}$ normalized by $f(0)=0$ and $f'(0)=1.$  For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where $r$ is a positive integer of order $2^{-r}$ $\left(0<2^{-r} \le \dfrac{1}{2}\right).$  By using  subordination, we propose a criterion for $f(z)\in S^* [a^{r},b^{r}].$ The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study  their regional transformations.  In addition, the properties of convexity for $f(z)\in A$ are examined.

### References

A. K. Bakhtin, I. V. Denega, Extremal decomposition of the complex plane with free poles, J. Math. Sci., 246, № 1, 1–17 (2020).

A. W. Goodman, Univalent functions, vol. I, Polygonal Publ. House, Washington, New Jersey (1983).

H. M. Srivastava, S. Owa (Editors), Current topics in analytic function theory, World Sci. Publ. Co., Singapore etc. (1992).

H. Liu, Sufficient conditions for certain analytic functions, Scholars J. Eng. and Technology (SJET), 2(2B), 243–246 (2014).

H. S. Kasana, Complex variables theory and applications, Prentice-Hall of İndia Private Limited, New Delhi (2005).

I. S. Jack, Functions starlike and convex of order $alpha$, J. London Math. Soc. (2), 3, 469–474 (1971).

M. Nunokawa, M. Aydogan, K. Kuroki, I. Yildiz, S. Owa, Some properties concerning close-to-convexity of certain analytic functions, J. Inequal. and Appl., 245, 1–10 (2012).

M. Orddovic, S. Owa, On certain properties for some classes of starlike functions, J. Math. Anal. and Appl., 145, 357–364 (1990).

N. Tuneski, Some results on starlike and convex function, Appl. Anal. and Discrete Math., 1, 293–298 (2007).

P. L. Duren, Univalent functions, Grundlehren math. Wiss., Bd. 259, Springer-Verlag, New York etc. (1983).

S. S. Miller, P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. and Appl., 65, 289–305 (1978).

S. S. Miller, P. T. Mocanu, On some classes of first order differential subordinations, Michigan Math. J., 32, 185–195 (1985).

Published
25.07.2023
How to Cite
ŞahinH., and Yildiz İsmet. “Determination of Some Properties of Starlike and Close-to-Convex Functions According to Subordinate Conditions With Convexity of a Certain Analytic Function”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 7, July 2023, pp. 995 - 1008, doi:10.37863/umzh.v75i7.7214.
Issue
Section
Research articles