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


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.


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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.
Research articles