Β
DSpace Repository
GEOMETRIC PROPERTIES OF INTEGRAL TRANSFORMS OF A SECOND ORDER DIFFERENTIAL INEQUALITY
- DSpace Home
- β
- MATHEMATICS
- β
- DOCTORAL THESES (Maths)
- β
- View Item
JavaScript is disabled for your browser. Some features of this site may not work without it.
GEOMETRIC PROPERTIES OF INTEGRAL TRANSFORMS OF A SECOND ORDER DIFFERENTIAL INEQUALITY
Davi, Satwanti
Date:
2014-12
Abstract:
Geometric function theory is a branch of classical complex analysis which deals with the
study of geometric properties of analytic functions. One of the broader area of current
research in geometric function theory is mainly concerned with the theory of univalent
functions. The most important fundamental result from the univalent function theory
is the Riemann mapping theorem, which states that every simply connected and proper
open subdomain of the complex plane can be mapped conformally onto the open unit
disk, upon an univalent function.
Let A be the class of all analytic functions f in the open unit disk D = fz 2 C : jzj < 1g,
which are normalized by the condition f(0) = f0(0) 1 = 0, and S, be the subclass of
A, consisting of all univalent functions in D. For the two normalized analytic functions
f1(z) = z+
X1
n=2
anzn and f2(z) = z+
X1
n=2
bnzn de ned in D, the convolution or Hadamard
product, denoted by ` ', is given by
(f1 f2)(z) = z +
X1
n=2
anbnzn; z 2 D:
In order to investigate the geometric properties of the analytic functions, several techniques
are available in the literature. Duality technique helps in solving complicated
problems very easily and also provides sharp results. For the most fundamental results
from the theory of duality that are used in thesis are from the monograph of S.
Ruscheweyh [116]. One of the application of this technique for convolution in connection
with an integral operator, was implemented by R. Fournier and S. Ruscheweyh in [36].
Let (t) : [0; 1] ! R be a non-negative integrable function which satisfy the normalization
condition
Z 1
0
(t)dt = 1. For f 2 A, consider the generalized integral transform
i
de ned by
V
(f)(z) =
Z 1
0
(t)
f(tz)
t
dt
!1=
; > 0 and z 2 D
where z or f represents only the principle part. This non-linear integral operator was
considered in the work of A. Ebadian et al. [34]. For the case = 1, this operator was
studied by R. Fournier and S. Ruscheweyh [36] and later by several authors, for di erent
choices of (t) (see [7, 9, 15, 138]). Note that V (f)(z) : V 1
(f)(z).
The analytic characterization of few of the important subclasses of the class S are given
below:
S ( ) =
f 2 A : Re
zf0(z)
f(z)
> ; 0 < 1; z 2 D
;
C( ) =
f 2 A : Re
1 +
zf00(z)
f0(z)
> ; 0 < 1; z 2 D
;
K( ) =
f 2A : Re
ei zf0(z)
g(z)
> ; 2 R; 0 < 1; g 2 S and z 2 D
and
C ( ) =
f 2 A : Re
1
1
zf0(z)
f(z)
+
1
1 +
zf00(z)
f0(z)
> ;
> 0; 0 < 1; z 2 D
o
;
where S ( ), C( ), K( ) and C ( ) are the class of starlike functions, convex functions,
close-to-convex functions and 1= - convex functions of order , respectively. Note that
S (0) S , C(0) C and K(0) K, respectively are the well known classes of starlike
functions, convex functions and close-to-convex functions that have rich geometric
properties.
The Pascu class denoted byM( ; ), 0 < 1, and de ned in D, is the hybridization of
two important subclasses: the class of starlike functions and the class of convex functions
of S. Analytically, this class is de ned as
M( ; ) =
f 2 A : Re
z(zf0(z))0 + (1 )zf0(z)
zf0(z) + (1 )f(z)
> ; 0 1; z 2 D
:
ii
Note that M(0; ) M( ).
One of the main objective of the thesis is to generalize and unify both the function class
P
( ; ), de ned in D and is given as
P
( ; )=
(
f 2A : Re ei
(1
)
f
z
+
f
z
zf0
f
!
> 0; 2 R
)
and the integral operator V
explained before, so as to de ne new functionals involving
the admissible functions and study the univalency and other geometric properties, using
duality technique. For this purpose, a more generalized class of P
( ; ) is de ned in
this thesis as under:
For 0, 0 < 1, > 0 and
0, let
W
( ;
) =
(
f 2A : Re ei
(1 +2
)
f
z
+
3
+
1
1
zf0
f
+
1
1+
zf00
f0
f
z
zf0
f
!
> 0; z 2 D; 2 R
)
:
Using this class and the integral operator V
(f)(z), the objective of the thesis is addressed
with several interesting applications. In the course of this study, interesting
analysis are made to study various consequences of W ( ;
) : W1
( ;
) as well.
The summary of all the seven chapters incorporated in this thesis is as follows.
Chapter 1 consist of fundamental de nitions, elementary concepts and few of the
existing literature on the geometric function theory that are used in the sequel. This
chapter constitutes the structures for the remaining chapters.
In Chapter 2, the conditions between the parameter and the function (t) are obtained,
under which the integral operator V maps the function from the analytic class
W ( ;
) into the class M( ) : M(0; ). For particular choices of (t), interesting
applications related to various well-known integral operators are obtained. The study is
also explored with the generalized linear operator, the operator formed with the convex
combination of z and V , which was de ned by R. M. Ali and V. Singh [9]. These results
are further extended for the class M( ; ) and are explained in Chapter 3.
iii
The admissible and su cient conditions on (t) are examined in Chapter 3, so that
the integral transforms V carries the function from the class W ( ;
) to M( ; ).
Various applications related to known integral operators are discussed. Clear analysis is
provided to explain the di erences and similarities of the results obtained in Chapter
2 and particular cases of Chapter 3.
The study of convex hull and extreme points for the class W (1; 0) was pursued by D.
J. Hallenbeck [49], and later H. Silverman [124] analysed for the analytic functions class
W (3; 1); and more recently, in [39, 74, 141].
De nition 1. An extreme point of a set E is a point of E that cannot be written as a
proper convex combination of any two points of E.
Chapter 4 provides the extreme points and sharp coe cient bounds for the class
W ( ;
). We also nd the estimates on , that would ensure functions in W ( ;
) are
starlike. When = 0, a sharp radius of univalence is obtained for the class W0( ;
).
De nition 2. The Schwarzian derivative (Sf (z)) of the analytic and locally univalent
function f in the domain D is de ned by the expression
Sf (z) =
f00(z)
f0(z)
0
1
2
f00(z)
f0(z)
2
:
It acts as an important tool in obtaining the su cient condition for univalence for the
locally univalent analytic function.
The su cient conditions in terms of the Schwarzian derivative and the second Taylor
coe cient is attained in Chapter 4 for functions belonging to the class W0( ;
).
In Chapter 5, the necessary and su cient condition is presented for starlikeness of
the generalized and non-linear integral transforms V
for the function belonging to the
analytic functions class W
( ;
). Applications associated with the well-known integral
operators are also discussed.
Chapter 6 deals with the admissible and su cient conditions on the parameter and
the function (t) so that the generalized and non-linear integral operator V
can be
mapped to the function from the class W
( ;
) into C ( ). Interesting consequences
corresponding to the known integral operators are examined.
iv
Chapter 7 determines the conditions between the parameters 1, 1,
1, 2, 2 and
2,
that carries the function from the class W
1( 1;
1) into W
2( 2;
2) for the generalized
and non-linear integral operator V
. We also study the number of applications for
speci c choices of (t) and the comparison is made with the existing results.
v
Files in this item
This item appears in the following Collection(s)
-
DOCTORAL THESES (Maths) [271]