ON BAZILEVIC FUNCTIONS

Let B(β) be the class of Bazilevic functions of type β(β>0). A function f ϵ B(β) if it is analytic in the unit disc E and Rezf′(z)f1−β(z)gβ(z)>0, where g is a starlike function. We generalize the class B(β) by taking g to be a function of radius rotation at most kπ(k≥2). Archlength, difference of coefficient, Hankel determinant and some other problems are solved for this generalized class. For k=2, we obtain some of these results for the class B(β) of Bazilevic functions of type β.


INTRODUCTION
. Bazilevic [i] introduced a class of analytic function f defined by the following relation.(i.i) where a is real, B > O, Re h(z)>O and g belongs to the class S of starlike functions.
Such functions, he showed, are univalent [I].With a=O in (I.i), we have for zeE Re zf' (z) > 0 (1.2) fl-8(z)gS(z This class of Bazilevic functions of type B was considered in [2].We de- note this class of functions by B(B).We notice that if B =i in (1.2), we have the class K of close-to-convex functions.
We need the following defi- nations.

Definition I.i
A function f analytic in E belongs to the vlass V k of functions with bounded boundary rotation, if f(0) 0, f'(0) I, f'(z) # 0, such that for z 9e eE, 0<r<l (zf' (z)) R e f"-(z/ d @ < k k>2 (1.3) For k=2, we obtain the class C of convex functions.
Definition 1.2 Let f be analytic in E and f(O)=O, f'(0)=l.Then f is said to belong  We now give the following generalized form of the class B(8).

Definition 1.3
Let f be analytic in E and f(0)=l, f'(O)=l.Then f belongs to the class Bk(8) 8>0 if there exists a geRk; k>2_ such that zf (z) We notice that, when 8=1, Bk(1) T duced and discussed in [6].Also B of close-to-convex functions.We shall give here the results needed to prove our main theorems in the preceeding section.
Lemma 2.1 [3].zeE Let feV k.IH(z) 12 -< l-(k2-1)r BAZILEVIC FUNCTIONS This result is known [6] and, for k=2, we obtain Pommerenke's result [7]   for functions of positive real parts Lemma 2. 3 Let S be univalent in E. Then: i (i) there exists a z I with ,IZll =r such that for all z, Izl =r 2 2r IZ-Zll IS l(z) I<_ 2 see [8]   (2.4) see [9]   (2.5) (l+r) (l-r) z+ Y, a z we define for j>l, n=2 n Then, with A0(n,zl,f) an; Let f be analytic in E and let the Hankel determinant of f be defined by (26).Then, writing Aj=Aj(n,zl,f), we have Lemma 2.5 With z q-2(n+q) A (n+2q-3) Let N and D be anlytic in E, N(0)--D(O) and D maps E onto many shee- ted region which is starlike with respect to the origin.
Let f be as defined in theorem 3.2.
Ilan+l I-la II 0(z)M n where 0(i) depends only on k and B.

I
Hence h is convex and thus starlike in IzI<r 0. The (-i+) valency foliows from the argument principle.Then g is starlike for IzI< r 0, where r is given by (3.4).
PROOF:Let GER k and let g be defined as in theorem 3.7.
Then 8 is starlike for JzJ<r 0' where r0 is given by (3.4) Now, from (3.5), we obtain z 1 Thus using lemma 2.6, we obtain the desired result that fgB(B) for Izl<r O, where r 0 is given by (3.4).

PROOF
1)-1Cl(8,k) is a constant depending only upon k and 8 yields )-i <__C(k,B)M I-(r) (r) whre C(k,B) is a constant depending only on k and 8.we obtain the required result.