Subordination And Convolution Of Multivalent Functions And Starlikeness Of Integral Transforms
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Date
2012-01
Authors
Badghaish, Abeer Omar
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Abstract
This thesis deals with analytic functions as well as multivalent functions de-
ned on the unit disk U. In most cases, these functions are assumed to be normalized,
either of the form
f(z) = z +
1X
k=2
akzk;
or
f(z) = zp +
1X
k=1
ak+pzk+p;
p a xed positive integer. Let A be the class of functions f with the rst normalization,
while Ap consists of functions f with the latter normalization. Five
research problems are discussed in this work.
First, let f(q) denote the q-th derivative of a function f 2 Ap. Using the theory
of di erential subordination, su cient conditions are obtained for the following
di erential chain to hold:
f(q)(z)
(p; q)zpô€€€q Q(z); or
zf(q+1)(z)
f(q)(z)
ô€€€ p + q + 1 Q(z):
Here Q is an appropriate superordinate function, (p; q) = p!=(p ô€€€ q)!, and
denotes subordination. As important consequences, several criteria for univalence
and convexity are obtained for the case p = q = 1:
The notion of starlikeness with respect to nô€€€ply points is also generalized to the case of multivalent functions. These are functions f 2 Ap satisfying
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Keywords
Analytic functions