NEW SUBCLASS OF STARLIKE FUNCTIONS OF COMPLEX ORDER

Abstract

The aim of the present paper is to investigate a new subclass
of starlike functions of complex order, b 6= 0. Let f(z) = z + a2z
2 + ... be
an analytic function in the unit disc D = { z| |z| < 1} which satisfies
1 + 1
b

z
f
0
(z)
f(z) − 1

=
1+Aω(z)
1+Bω(z)
,for some ω ∈ Ω and for all z ∈ D.
Then f is called a Janowski starlike function of complex order b, where
A and B are complex numbers such that Re(1 − AB¯) ≥ |A − B|, im(1 −
AB¯) < |A − B|, |B| < 1, and ω(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.