Differential subordination and coefficients problems of certain analytic functions
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Date
2014-09
Authors
Supramaniam, Shamani
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Abstract
Let A be the class of normalized analytic functions f on the unit disk D, in
the form f(z) = z+
P1n
=2 anzn: A function f in A is univalent if it is a one-to-one
mapping. This thesis discussed ¯ve research problems.
A function f 2 A is said to be bi-univalent in D if both f and its inverse
f¡1 are univalent in D. Estimates on the initial coe±cients, ja2j and ja3j, of
bi-univalent functions f are investigated when f and f¡1 respectively belong to
some subclasses of univalent functions. Next, the bounds for the second Hankel
determinant H2(2) = a2a4 ¡ a23
of analytic function f for which zf0(z)=f(z) and
1 + zf00(z)=f0(z) is subordinate to certain analytic function are obtained.
Motivated by the earlier work on second order di®erential subordination for
analytic functions with ¯xed initial coe±cient, the su±cient conditions for star-
likeness and univalence for a subclass of functions with ¯xed second coe±cient
are obtained. Then, without ¯xing the second coe±cient, the su±cient condition
for convexity of these functions satisfying certain second order and third order
di®erential inequalities are determined.
Lastly, the close-to-convexity and starlikeness of a subclass of multivalent func-
tions are investigated.
A few aspects of problems in univalent function theory is discussed in this
thesis and some interesting results are obtained.
Description
Keywords
Subordination , Coefficient