UNIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS

Theorem 2.1. Let f(z) = z + E:'=2 anzn be in S. If E:'=2(n - ,B)lanl ~ (3- 1 then f E M((3). Received December 29, 1992. 1991 Mathematics Subject Classification. Primary 30C45. Key words and phrases. Univalent, starlike, convex. 225

Corollary 2.2. Let f(z) = z+ E:'=2 anzn be in S. If E:'=2 n(n-/3)lanl ~ /3- l then f E L(/3).

Theorem 2.3. A function f(z) = z + E:'=2 lanlzn is in V(/3) if and only if E:'=in - /3)lanl ~ /3 - l.

Corollary 2.4. A function f(z) = z + E:=2 lanlzn is in U(/3) if and only if E:=2 n(n - /3)lanl ~ /3 - 1.

Theorem 2.3 enables us to prove the following

Theorem 3.1 If f E V(/3) then /3-1 . · /3-1 r- 2_13r2 ~ lf(z)I ~r+ 2_13r2 (lzl = r) with equality for f(z) = z + t:Jz2

Corollary 3.2. If f E U(/3) then /3-1 2 /3-1 2 r - _,_ ~J ~ lf(z)I ~ r + _ ,_ _, r (lzl = r) with equality for f(z) = z + 2&-=:.1>z2 (z = ±r)

Theorem 3.3. · The disk lzl < 1 is· mapped on to a domain that contains the disk lwl < (3 - 2/3)/(2 - /3) by any f E V(/3) and on to a domain that contains the disk lwl < (5 -· 3/3)/2(2 - /3) by any f E U(/3). The theorem is sharp for the extremal functions z + t~J z2 E V (/3) and z + 2'2-=:.~> z2 E U (/3).

Theorem 3.4. If f E V(/3) then 1- 2(/3- l)r < 1/'(z)I < 1 + 2(/3- l)r 2-/3 - - 2-/3 (lzl = r)

Corollary 3.5. If f E U(f3) then 8-1 /3-1 1 - - · -r < IJ'(z)I < 1 + --r 2-(3 - - 2-(3 (lzl = r). Equality holds for f(z) = z + "fl,-~, z2 (z = ±r) 4. Order of Starlikeness and Convexity

Theorem 4.1. If J E V(f3) then J E V*((4 - 3/3)/(3 - 2/3))

Corollary 4.2. V(/3) c V(4/3) CV*.

Corollary 4.3. If f E U(/3) then f E VK((4 - 3/3)/(3 - 2/3))

Corollary 4.4. U(/3) C U(4/3) C VK. The above corollary is comparable t6 the following results of S. Ozaki [1] and R.Singh and S.Singh [3], for wider class of functions. Theorem A. [l]. If f(z) = z + E:'=2 anzn is analytic in E and satisfies Re (l + zf"(z)/ f'(z)) < 3/2 then f is univalent in E. Theorem B. [3]. If f(z) = z + E:'=2 anzn is analytice in E and satisfies Re(l + zf"(z)/ f'(z)) < 3/2 then f is starlike in E.

Theorem 4.5. If J E U(/3) then f E V(2/(3 - /3)).

Corollary 4.6. U(4/3) C V(6/5). From Corollary 4.6 and Theorem 4.1, we have

Corollary 4.7. U(4/3) C V*(2/3). Since Theorem 4.5 is true even if 1 < /3 ~ 3/2 the following Corollary is obtained.

Corollary 4.8. If f(z) = z + E:'=2 lanlzn E V, satisfies Re(l + zf"(z)/ f'(z)) < 3/2 then Re zf'(z)/ f(z) < 4/3 i.e. f E V(4/3). 5. Extreme Points In view of Theorem 2.3 the class V(/3) is closed under convex linear combinations. We shall determine the extreme points of V(/3).

Theorem 5.1. Let fi(z) = z and fn(z) = z + ~=~zn,n = 2,3,.. .. Then f E V(/3) if and only if it can be expressed in the form f(z) = E:'=1 >..nfn(z).

Corollary 5.2. The extreme points of V(/3) are the functions fn(z), n = 1, 2, ...

Corollary 5.3. The extreme points of U(/3) are the functions fi(z) = z and fn(z) = Z + ..,f;.--:._ 1R,zn,n = 2,3, ... References [1] S. Ozaki, "On the theory of multivalent functions II," Science Reports of the Tokyo Bunrika Daigaku Section A, 4(1941), 45-87. [2) H. Silverman, "Univalent functions with negative Coefficients," Proc. Amer, Math. Soc., 50 (1975), 109-115. [3) R. Singh and S. Singh, "Some sufficient conditions for Univalence and Starlikeness," Colloqu.iu.m Mathematicum, XLVII (1982)