V23N4-P355-362_OCR

Lemma 2.1. Let a EC, la-11 < b :S Re(a) and p(z) be analytic in E with p(0) = 1. Then IP(z) - a I < b, z E E, if and only if, there exists a function w analytic in E satisfying w(O) = 0, lw(z)I < l for z E E such that p(z) = 1 + !l w(z) ｀＇六 ，丶, z EE, where A= (b2 - lal2 + a)/u and JJ = (I - a)/&. The proof of this lemma follows by an a.pplication of Schwarz's lemma as in Rotaru [6J. From now on A and 13 will be a.s in lemma 2. 1 and also c = (b2 - la - 庄）/b. Next we have the well known Jack's le

Theorem 2.1. Let f(z) E S"'_(o·,a,b), o. > (4/c) llm(a)I. Then f(z) E S*(a,b).

Corollary 2.1. Let f(z) E S*(a,a,b), ex> 0. Then J(z) E S*(a,b). The above corollary follows by taking a = 五in theo_rem 2.1.

Corollary 2.2. Let a> 0, b > l/2 and f(z) E S*(a,b,b). Then f(z) E S*(b,b). The above result is obtained if, in corollary 2.1, we take a= b, b > 1/2.

corollary 2 .1.

Corollary 2.3. Leto·> 0, 0~p < 1, 0 < (3~l and f E V with J(z)J'(z)/z :j; 0 for z in E. Then I (zf'(z)/f(z)) - 1 翊((zf'(z)/J(z)) - p) - ((zf'(z)/J(z)) - 1) I < 1, for z in E. whenever I J(o:,f(z))-1 2{3(J(a,f(z)) - p)- (J(o:, f(z)) - 1 I < 1, for z in E, where J(a, f(z)) = (1- o:)(zf'(z)/f(z)) + o:(zf'(z))'/f'(z) and a a non-negative real number.

Theorem 2.2. For (4/ c) I/ m(a)I < 且< n, S*(a:,a,b) c S*(趴a, b).

Corollary 2.4. For a~l > (4/c) llm(a)I, S*(a,a,b) C K(a,b). 3. In this section, we obtain an in1portant integral representation for the elements of S*(a,a,b).

Theorem 3.1. Let f(z) E S*(o:,a,b), o: > (4/c) llm(a)I, and if for (4/c) jJm(a)I < (J <awe choose the branch of [zf'(z)/f(z)]P which is equal to l when z = 0, then the function F13(z) = f(z) [zf'(z)/f(z)]P is in S*(a,b).

Theorem 3.2. Let F(z) E S*(a,b) and a > (4/c) IIm(a)I. Then f(z) defined by (3.1) belongs to S*(a,a,b).