Convolution and coefficient problems for multivalent functions defined by subordination
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Date
2009
Authors
Supramaniam, Shamani
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Let C be the complex plane and U := {z 2 C : |z| < 1} 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 f analytic in the open unit disk U := {z 2 C : |z| < 1},
and normalized by f(0) = 0, and f0(0) = 1. A function f 2 A has the Taylor series
expansion of the form
f(z) = z +
X1
n=2
anzn (z 2 U).
Let Ap (p 2 N) be the class of all analytic functions of the form
f(z) = zp +
X1
n=p+1
anzn
with A := A1.
Consider two functions
f(z) = zp + ap+1zp+1 + · · · and g(z) = zp + bp+1zp+1 + · · ·
in Ap. The Hadamard product (or convolution) of f and g is the function f g
defined by
(f g)(z) = zp +
X1
n=p+1
anbnzn.
In Chapter 1, the general classes of multi-valent starlike, convex, close-to-convex and
quasi-convex functions are introduced. These classes provide a unified treatment to
various known subclasses. Inclusion and convolution properties are derived using the
methods of convex hull and differential subordination.
In Chapter 2, bounds for the Fekete-Szeg¨o coefficient functional associated with
the k-th root transform [f(zk)]1/k of normalized analytic functions f defined on U
are derived for the following classes of functions:
Rb(') :=
f 2 A : 1 +
1
b
(f0(z) − 1) '(z)
,
S ( , ') :=
f 2 A :
zf0(z)
f(z)
+
z2f00(z)
f(z)
'(z)
,
L( , ') :=
(
f 2 A :
zf0(z)
f(z)
1 +
zf00(z)
f0(z)
1−
'(z)
)
,
M( , ') :=
f 2 A : (1 − )
zf0(z)
f(z)
+
1 +
zf00(z)
f0(z)
'(z)
,
where b 2 C \ {0} and 0. A similar problem is investigated for functions z/f(z)
when f belongs to a certain class of functions.
In Chapter 3, some subclasses of meromorphic univalent functions in the unit
disk U are extended. Let U(p) denote the class of normalized univalent meromorphic
functions f in U with a simple pole at z = p, p > 0. Let be a function with
positive real part on U, (0) = 1, 0(0) > 0, which maps U onto a region starlike
with respect to 1 and which is symmetric with respect to the real axis. The class
P (p,w0, ) consists of functions f 2 U(p) meromorphic starlike with respect to w0
and satisfying
−
zf0(z)
f(z) − w0
+
p
z − p
−
pz
1 − pz
(z).
The class
P
(p, ) consists of functions f 2 U(p) meromorphic convex and satisfying
−
1 + z
f00(z)
f0(z)
+
2p
z − p
−
2pz
1 − pz
(z).
The bounds for w0 and some initial coefficients of f in
P (p,w0, ) and
P
(p, ) are
obtained.
Description
Master
Keywords
Mathematical science , Multivalent functions , Subordination