### New subclass of starlike functions of complex order

**Yasar Polatoglu, H. Esra Ozkan**

**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.