Results & Lemmas (5)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
THEOREM 1.
THEOREM 1. Let G(z) be convex conformal map of D, C(0) = p, f rz (2.1) F(z) = s exp 0 dx. Let (2.2) f z) = zl be analytic in
THEOREM 1. Let G(z) be convex conformal map of D , C(0) = p , f rz (2.1) F(z) = s exp 0 dx . Let (2.2) f{z) = zl be analytic in
COROLLARY 1.
COROLLARY 1. Let f(z) be analytic in D and of the form (2.2). Then (2.8) Re > 3 » 0, z € D, for certain a. € (-(ir/2), ir/2j, p cos a > &,…
COROLLARY 1. Let f(z) be analytic in D and of the form (2.2). Then (2.8) Re\eia > 3 » 0 , z € D , for certain a. € (-(ir/2), ir/2j , p cos a > & , if and only if (2//V)(pcosa-B)exp(-ia) (2 9) (t/s)p it/a; l-t»z 1-8 z for all
COROLLARY 2.
COROLLARY 2. Let fiz) be analytic in D and of the form ( 2. 2 ). Then R e e W aC^f)| > 6 > 0, z € D, if and only if t € (0, 2 cos a), where…
COROLLARY 2. Let fiz) be analytic in D and of the form ( 2 . 2 ) . Then R e e W aC^f)| > 6 > 0 , z € D , if and only if t € (0, 2 cos a) , where \ (2./N) cosa( g-pcosa) :is bound is sharp and one can easily check that Fit, a, @, p, A) < 1 at least for all t € (0, cos a) .
THEOREM 2.
THEOREM 2. Let G[z), f z), <p(w) be as in Definition 1 and let (2.11) EM = » and F(z) as in (2.1). Then f(z) is p-valent N-ip-like with…
THEOREM 2. Let G[z) , f{z) , <p(w) be as in Definition 1 and let (2.11) EM = » and F(z) as in (2.1). Then f(z) is p-valent N-ip-like with respect to G if and only if, for all \s\ 5 1 3 | t | 5 1 , R\f{tz)) N F(tz) • (2 12)
THEOREM 3.
THEOREM 3. Let /(z) be analytic and p-valent in D. Then, for (3.1) f[tzN) <s f z), z € D.
THEOREM 3. Let /(z) be analytic and p-valent in D . Then, for (3.1) f[tzN) <s f{z) , z € D .