ANALYTIC FUNCTIONS OF NON-BAZILEVIČ TYPE AND STARLIKENESS

Two classes ̄n(μ,α,λ) and ̄n(μ,α,λ) of analytic functions which are not Bazilevǐc type in the open unit disk U are introduced. The object of the present paper is to consider the starlikeness of functions belonging to the classes ̄n(μ,α,λ) and ̄n(μ,α,λ). 2000 Mathematics Subject Classification. 30C45.

a j z j analytic in U . ( Also, we need the following notations and definitions.Let be the class of starlike functions (with respect to the origin) in U, and let be the subclass of * .Further we define which is the subclass of Bazilevič class of univalent functions (cf.[1]).Ponnusamy [8] has considered the starlikeness and other properties of functions f (z) in the class Ꮾ(µ, λ).For negative µ, that is, for −1 < µ < 0, which is better to write (with 0 < µ < 1) in the form we obtain the class which was considered earlier by Obradović [3,4], Obradović and Owa [5], and Obradović and Tuneski [6].
For the limit case µ = 0, this class becomes the classλ .When µ = 1, this class becomes the class of univalent functions f (z) satisfying which was studied by Ozaki and Nunokawa [7].
Next, for α = (α 1 ,α 2 ,...,α k ) ∈ R k with α j ∈ R, we define the operator D α by and, by virtue of the operator D α , the subclasses Samaris [9] has investigated the appropriate classes for the case (1.4), and has obtained results which are stronger than those given earlier and in several cases sharp ones.By using the method by Samaris [9], we will generalize some results given in [3,4], and we will obtain some new results.We also note that we cannot directly apply some nice estimates given by Samaris [9].

Starlikeness of the classes
Ꮾn (µ,α,λ) and ᏼn (µ,α,λ).Our first result is contained in the following theorem.Theorem 2.1.Let Ꮾn (µ,α,λ) be the class defined by (1.8) for which Proof.From we obtain (2.2) 2), we have ( From (2.4), we easily obtain that and from here from which the conclusion of the theorem easily follows.
Proof.For t 1 ∈ R and t 2 ∈ R, with the lemma by Fournier [2], there exists a sequence of functions φ k (z) analytic in the closed unit disk Ū such that (2.12) Since lim k→∞ w k (z) = z n e it 1 and lim k→∞ w k (1) = e it 2 , we see that (2.13) For t 1 = 0 and t 2 = 0, we get
If we choose t 1 = q 1 + π/2, t 2 = π/2 − q 2 , then we obtain and therefore (2.16) From cos(q 1 + q 2 ) 0 or sin 2 q 1 + sin 2 q 2 − 1 0, we have the statement (ii) (one part) of the theorem.We note that the "if" part of the theorem follows from the result of Theorem 2.1 and from the result given by Obradović and Owa [5].

.18)
Proof.The proof of this theorem is similar to that of Theorem 2.1.By virtue of (2.20) From the last relation, we see that The statement of this theorem follows from the above inequality.
Finally we give the following example of the theorem.

1 .
Introduction.Let U = {z ∈ C : |z| < 1} denote the open unit disk in the complex plane C. For n 1, we define 1, if f (z) ∈ Ꮽ n satisfies