Abstract
Let /(z) = z+ L akzk be analytic in the unit disc E = {z: lzl < l}.
k=2
We wish to maximize la3 - ua~ I over certain classes of analytic functions defined
by convex subordination. This paper is concerned with the solution of the above
extremal problem over certain classes of univalent analytic functions.
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 2.1.
Theorem 2.1. If f E M(a; A, B), then (i) forµ complex (A - B) I a3 - µa~ I ~ 2(1 + 2a)' (A - B)2. ",'1 __L ~,2 Iµ -, I, Iµ -, I~ v, Iµ -, I…
Theorem 2.1. If f E M(a; A, B), then (i) forµ complex { (A - B) I a3 - µa~ I ~ 2(1 + 2a)' (A - B)2 . " ,'1 __L ~,2 Iµ - , I, Iµ - , I~ v, Iµ - , I 2:: v; (2·.1) (2.2) and {ii) for µ real,
Theorem 2.2.
Theorem 2.2. Let f E H(a; A, B), then (i) forµ complex, (A- B) 3(1+a)' (A - B)2 A I /1 - µ I, 4 I,1 - µ I~ "'' · )(A - B)' 4. I r - µ I >
Theorem 2.2. Let f E H(a; A, B), then (i) forµ complex, { (A- B) 3(1+a)' (A - B)2 A I /1 - µ I, 4 I ,1 - µ I~ "'' · )(A - B)' 4 . I r - µ I >
Theorem 2.3.
Theorem 2.3. Let f E B1(a;A,B), then (i) for any complex numberµ, (A - B) I a3 - pa~ I ~ 2 + a, I r2 - µ I (A - B)2 1", - ? I r2 - µ I,…
Theorem 2.3. Let f E B1(a;A,B), then (i) for any complex numberµ, { (A - B) I a3 - pa~ I ~ 2 + a , I r2 - µ I (A - B)2 1" , - \? I r2 - µ I, (l+a)2 ... < Ir. 1 '\ I ,1 r"I, 1 I I
Theorem 2.4.
Theorem 2.4. If f E F(a; A, B), then (i) forµ complex, (A-B) 3(1 + 2a)' I a3 - µal I:S '.~ ~ ~i: Iµ +,. I, and 4(1+a)2 lµ+,a I~::…
Theorem 2.4. If f E F(a; A, B), then (i) forµ complex, (A-B) { 3(1 + 2a)' I a3 - µal I :S '.~ ~ ~i: Iµ + ,. I, and 4(1+a)2 lµ+,a I~ ::\(A_R)(J-1-?,..\, 4(1+a)2 • I JJ, + 13 I 2:: ::\( A - R \( 1 -l- ? I'\') ' (2.27) (2.28)
Theorem 2.5.
Theorem 2.5. If f E G(a;A, B)1 then (i) for any complex numberµ, I a3 - µa~ I ~ (A - B) (1 + 2a)' (A - B)2 /1 I -. ? Iµ + 14 I, (1 + a):.i…
Theorem 2.5. If f E G(a;A, B)1 then (i) for any complex numberµ, I a3 - µa~ I ~ { (A - B) (1 + 2a)' (A - B)2 /1 I -.\? Iµ + 14 I, (1 + a):.i lµ+,41~ fA-RV1..l..?"\' (1 + a):.i Jµ+,412:: (,4_R\(1..L?~\, (2.38) (2.39)
Function classes studied:
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