Journal of Prime Research in Mathematics

# New subclass of starlike functions of complex order

Yasar Polatoglu
Department of Mathematics and Computer Sciences, Kultur University, Turkey.
H. Esra Ozkan
Department of Mathematics and Computer Sciences, Kultur University, Turkey

$$^{1}$$Corresponding Author: y.polatoglu@iku.edu.tr

### Abstract

The aim of the present paper is to investigate a new subclass of starlike functions of complex order $$b\neq 0$$. Let $$f(z)=z+a_{2}z^{2}+…$$ be an analytic function in the unit disc $$D=\{z| |z|<1\}$$ which satisfies $$1+\frac{1}{b}(z\frac{f'(z)}{f(z)}-1)=\frac{1+A\omega z}{1+B\omega z}$$, for some $$\omega \in \Omega$$ and for all $$z \in D$$. Then f is called a Janowski starlike function of complex order b, where A and B are complex numbers such that $$Re(1-A\overline{B})\geq |A-B|, im(1-A\overline{B}<|A-B|, |B|<1$$ and $$\omega(z)$$ ) is a Schwarz function in the unit disc D [1], [10], [12]. The class of these functions is denoted by $$S^{∗}(A, B, b)$$. In this paper we will give the representation theorem, distortion theorem, two point distortion theorem, Koebe domain under the montel normalization, and coefficient inequality for this class.

#### Keywords:

Starlike, distortion, Koebe, Montel normalization, coefficient.