PENCIRIAN FUNGSI BAK-BINTANG PARABOLIK DAN FUNGSI CEMBUNG SECARA SERAGAM TERITLAK
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Date
2000-03
Authors
KEONG, LEE SEE
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Abstract
Let U = { z : I z I < 1 } be the open unit disk in the complex plane and S be the class of analytic
=
univalent functions f(z) = z + L, anzn normalized so that f(O) = f' (0) -1 = 0. Let C
n=2
denote the class of functions g E S which maps U onto a convex domain. Also, let s* denote
the class of functions f E S which maps U onto a domain starlike with respect to the origin.
A. W. Goodman introduced the class of uniformly convex functions CSS consisting of
functions g E C with the property that for every circular arc r contained in U, with center also
in U, the image arc g(y) is a convex arc. Similarly, Goodman defined the class of uniformly
starlike functions BBS consisting of functions f E S • with the property that for every circular
arc r contained in U, with center also in U, the image arc j(y) is a starlike arc with respect to
the origin.
For the classes C and s*, it is known that g E C if and only if f = zg' E S *. One would
expect then that there would be a similar relation between CSS and BBS. However, this is not
true as proved by Goodman, in which he showed that a function f = zg' E BBS does not
necessarily imply that g E CSS . For functions f E BBS which actually satisfy the above
condition, F. RlZ\nning categorized these functions to be in the class S P, that is f = zg' E S P if
and only if g E CSS. Geometrically, the function f E S P will map U into a parabolic region
Q = { w = u + iv : v 2 < 2u - 1 } = { w: I w - 1 I< Ny w }
IX
R. M. Ali and V. Singh generalized the class S P by allowing the above parabolic region n to
vary so that the vertex will be located within the interval (0,1) on the real axis.
In this thesis, we will generalize the class CSS to a class CSS(p) and relate it with the
generalized class Sp(p). Some analytic properties of the generalized CSS(p) will be
obtained. We will also obtain the sharp bound for the Fekete- Szego functional I a 3 - pa/ I
~
for functions j(z) = Z + L anzn E S p (p).
n=2
Finally, we will study the generalized
hypergeometric functions and determine conditions for these functions to be in the class
CSS(p) and S P (p).
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Keywords
PENCIRIAN FUNGSI BAK-BINTANG PARABOLIK