GROWTH RESULTS FOR A SUBCLASS OF BAZILEVI FUNCTIONS

For 0, let B() be the class of regular normalized Bazilevi functions defined in the unit disc. Choosing the associated starlike function g(z) z gives a proper subclass BI() of B(). For B(), correct growth estimates in terms of the area function are unknown. Several results in this direction are given for BI(1/2).


INTRODUCTION.
Let S be the class of regular, normalized, univalent functions with power series Denote R, S*, K and B() the subclasses of S which are functions whose deri- vative has positive real part [8], starlike with respect to the orgin [9 p.221], close- to-convex [6] and Bazilevic of type [13] respectively.Following [13] we define f B(a), 0 to be the class of functions f regular and normalized in D, such that, there exist g e S* such that for z e D, Rc zf'(z) O.
(1.2) f(z) l-g (z)a Then if g(z) f(z), B() S* and B(1) K Let C(r) denote the closed curve which is the image of D under the mapping w f(z), L(r) be the length of C(r) and A(r) the area enclosed by the curve C(r) For f e S*, it was shown [7] that, where M(r) max f(z)l, and Hayman [4] gave an example to show that this estimate is best possible when f is bounded.In [14] this result was extended to starlike func- tions with A(r) A constant.A modification of this method also shows .thatfor f S*, e(r) 0(1) /A(r)(log _J__l l-r as r-I. (I 4) Thomas [14] also showed that (1.3) holds for the class K and for the class B(a), 0 a _< [13].It is apparently an open question that (1.4) is valid for f K or B(a).
Pommerenke [II] showed that if f S* then for n 2 nla C/A(In -)' I 6) n n The question as to whether (1.5) is valid for f K or B(a) is also apparently open.
In [12] the subclass Bl(a of B(a) consisting of those functions in B(a) for which g(z) _= z was considered and sharp estimates for the modules of the coefficients a2, a3, and a 4 were given.In [15] Thomas gave sharp estimates for the coefficients a in (i.i) when a I/N, N a positive integer n In this paper we shall be concerned with the class BI(1/2) and will use the method of Clunie and Keogh [I] to establish (1.5) and hence (1.6) and the method of Thomas [14]   to prove (1.4) and hence (1.3).The methods will in fact give results which are stronger for this subclass.
2. STATEMENTS OF MAIN RESULTS.
PROOF OF THEOREM I.
This means that the coefficient combination nan b2n-1 on the left hand side of (2.2) depends only on the coefficient combinations [2a (kak+b )z2k-1}w(z2) -1 k=n+l say.Squaring the moduli of both sides of (2.3) and integrating round Izl r, we obtain, using the fact that lw(z2)l < (2.5) Finally, it is easy to see that from the definition of BI(), F e R and so [8] for 2 n 2, Ib2n_ll -Z" Thus (2.5) gives nlanl o(I) + 0(I) /(A(I )) as n =.
This proves part (i) of Theorem I.
We note that (ii) is stronger that (i) and that we have proved (ii) and (iii) in the case when A(r) is finite.
The following extensions to Theorem support the above conjectures.

BI
h is harmonic in D Thus J2(r) _< 47 log l-r" pdOdp Combining the estimates for Jl(r) and J2(r) shows that 12(r) 0(i) /(A(r)) log (r) as r and the result is proved.COROLLARY i.Let f BI(1/2) then as n ,(i)  nlanl.. o(I) + 0(i) e(1 !)and the fact that Ib2k_l <-2k--I for k I, it follows that, ) _< (n=l nlan]2rn)1/2(n=l rn)1/2"n It follows trivially from(2.1)that e(r) 0(I) M(r) log -r as r and so (iii) follows at once on noting that M(r) 2 _< A(/r)_ log.l-r and on using lemma 2.REMARK.In view of Theoremand Corollary i, it is possible that for f