PRESTARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS

. The extreme points for prestarlike functions having negative coefficients are determined. Coefficient, distortion and radii of univalence, star-likeness, and convexity theorems are also obtained.

and is said to be in the class of functions conve_x o_.f a, denoted by K(a), if Re{l + zf"(z)/f'(z)} > (z U).
Further, let T and T* [a] denote the subclasses of S and S*(a), respectively, whose n elements can be expressed in the form f(z) z I ..lanlZ n=2 n The convolution or Hadamard product of two power series f(z) En=0 anZ and g(z) In=0 bn zn is defined as the power series (f'g)(z) In=0 anbn zn" An analytic function normalized by f(0) f' (0) 1 0 is said to be in the class of functions prestarlike of order , 0 < a < i, denoted by R if f*s S*() where s (z)   z/(-z) 2(-) The function s is the well-known extremal function for the class S*(a).In the sequel, we let n (k-2a) k=2 C(,n) (n-l) (n=2, 3, ), (I.I) so that s can be written in the form n sa(z) z + Zn=2 C(a,n)z Note that C(a,n) is a decreasing function of a with , a < 112 The class R was introduced by Ruscheweyh [3], who showed that a necessary and sufficient condition for f to be in R is that the functional satisfy G(=,z) Re G(e,z) > 1/2 (z U). (1.3) Since Sl(Z) z, we say that f is prestarlike of order 1 if and only if S*(I/2).In [3] it was Re{f(z)/z} > 1/2 (z U).Note that R 0 K(0) and RI/2 shown that Rs c R 8 for 0 < < 8 < I, which generalizes the well-known result that K(0) c S*(I/2).
In Section 2, we obtain a sufficient condition in terms of the modulus of the coefficients for a function to be in R and show that this condition is also nec- essary for the subclass R[] R n T. (1.4) In Section 3, we obtain the extreme points for the closed convex hull of R[] and use them to prove distortion and covering theorems.In Section 4, we determine the radii of univalence, starlikeness, and convexity for R[e].Finally, we find the smallest 8 8(e) for which T*[] c R[8], 0 < e < I.
2. COEFFICIENT INEQUALITIES FOR THE CLASS RIll.
We first obtain a relationship between the order of prestarlikeness of a function and the modulus of its coefficients.
REMARK.For the case 1 in the theorem and all subsequent results, the expression (n-)C(,n)/(l-) is taken to be 2, its limit as / i PROOF.An equivalent formulation of (1.3) is I/G(,z)-I < i, 0 !<_ i.
For s (z) which is bounded by 1 whenever (2.1) is satisfied.
The converse of Theorem 1 is also true for the class RIll, defined by (1.4).
Since a necessary and sufficient condition [4] for we obtain We now determine the extreme points of this class.

EXTREME POINTS OF R[].
For any compact family of analytic functions , it is well known that the real part of any continuous linear functional over is maximized (minimized) at one of the extreme points of the closed convex hull of .The solutions to several extremal problems in R[] follow easily from the extreme points for this class.In view of Theorem 2, we see that R[] is a closed convex family.Thus, the extreme points are obtained in THEOREM 3. Set fl(z) z and fn(Z) z (l-)zn/(n-)C(,n), n=2, 3, Then f RIll, 0 _< _< i, if and only if it can be expressed in the form f(z) Y.
As an immediate consequence of Theorem 3, we obtain distortion theorems for the class RIll.
COROLLARY.The disk zl < 1 is mapped onto a domain that contains the disk lwl < (3-2e)/(4-2e) for any f E R[], 0 < < i.The result is sharp, with extremal (1-a)nrn-1 (n-a) C'(a,n) is a decreasing function of n.In view of (3.I), the inequality g(a,r,n+l) < g(a,r,n Since, for n fixed, h is a decreasing function of r, we have h(,r,n) > h(a,l,n) (l-2a)n + (l-a)(l-2a) + a 0 for s < 1/2.Since h is also a decreasing function of s, it follows for r < 2/3 that h(,r,n REMARKS.I. Since h(l,r,2) 2 3r < 0 for r 2/3, we have g(l,r,2) < g(l,r,3).Thus the corollary will not be true for all when r > 2/3.
2. We next show that the corollary will not be true for all r when a I/2.For each a, 1/2 < a < I, we must find an r r(a) such that M(u,r) > (I/ (2-u))r.It suffices to show for n n(u) sufficiently large that Since C(s,n) / 0 for 1/2 and the right hand side of (3.2) is bounded below by (l-s)(2-) 0, the result follows.
The functions in R[] for 0 < = < 1/2 are starlike of a positive order.The bound is given in THEOREM I, it suffices to show that g(,n) is a decreasing function of n.
REMARK.For 0, Theorem 6 reduces to the known result [4] that When > 1/2, R[] S. We will show that gn(Z) z 2zn/n R[] S for n n() sufficiently large.
We next determine the largest disk in which R[] is univalent.THEOREM 7. The radius of univalence and starlikeness n= 2 nlanlr n-I <_ i.In view of Theorem 2, this is true if < r(n_)C(,n)] I/(n-1) r n(l-) L ] Hence, f is starlike for Izl <_ r().On the other hand, for some n we have n Thus f is not univalent (or starlike) for Izl <_ r, r > r().
COROLLARY.The radius of univalence and starlikeness for R[I] is i/ m 794.
Thus, for all , f in R[] is univalent and starlike when Izl < i'/.
REMARK.MacGregor showed [2] that the radius of univalence and starllkeness for R I is I/v .707.
We will now obtain the radius of convexity for R The result is sharp, with the extremal function of the form n-I which is bounded by I whenever En=2 a z n < I. From Theorem 2, this will hold whenever The result follows upon setting Izl r(a) in (4.1).
Using arguments similiar to those in the corollary to Theorem 7, we have the following COROLLARY.The radius of convexity for R [I]  is nonnegative for 0 < = < 1/2, the first inclusion is proved.Since C(7,n) / for y < i/2 and the right hand side of (4.2) is bounded, the in- equality is not true for n sufficiently large.
The inclusion relation T*[e I] c T*[2] for e I > e 2 shows that T*[] c R[I/2] for > 1/2.But for any e < 1 and y < 1/2, we will show that f