Supramaniam, Shamani
(2009)
Convolution And Coefficient Problems
For Multivalent Functions Defined By
Subordination.
Masters thesis, Universiti Sains Malaysia.
Abstract
Andaikan C satah kompleks, U = {z E C : Izl < I} cakera unit terbuka dalam
C dan H(U) kelas fungsi analisis dalam U. Andaikan juga A kelas fungsi analisis
1 dalam U yang ternormalkan dengan 1(0) = 0 dan 1'(0) = 1. Fungsi 1 E A
mempunyai siri Taylor berbentuk
00
l(z) = z + L anzn, (z E U).
n=2
Andaikan Ap (p EN) kelas fungsi analisis 1 berbentuk
00
1(z) = zP + L anzn, (z E U)
n=p+1
dengan A := AI.
Pertimbangkan dua fungsi
dalam Ap. Hasil darab Hadamard (atau konvolusi) untuk 1 dan 9 ialah fungsi 1 * 9
berbentuk
00
(J * g)(z) = zP + L anbnzn.
n=p+1
Let C be the complex plane and U := {z E C : Izl < I} be the open unit disk
in C and H(U) be the class of analytic functions defined in U. Also let A denote
the class of all functions I analytic in the open unit disk U := {z E C : Izl < I},
and normalized by 1(0) = 0, and 1'(0) = 1. A function I E A has the Taylor series
expansion of the form
00
I(z) = z + ~ (LnZn (z E U).
n=2
Let Ap (p EN) be the class of all analytic functions of the form
00
fez) = zP + ~ (LnZn
n=p+l
with A:= AI.
Consider two functions
in Ap. The Hadamard product (or convolution) of I and 9 is the function I * 9
defined by
00
(J * g)(z) = zP + ~ anbnzn
.
"=p+l
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