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

• Date: 2023
• Main Authors: Şahin, Hasan, Yildiz, İsmet
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2023
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/7214
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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.