0
‫بسم هللا الرحمن الرحيم‬
Al al-Bayt University
Faculty of Science
Department of Mathematics
Some Properties of Analytic Function of
Complex Order
By
Ala'a Mohammd Ibrahem Al-Shdifat
Supervisor
Dr. Basem Aref Frasin
2011
1
‫بسم هللا الرحمن الرحيم‬
Al al-Bayt University
Faculty of Science
Department of Mathematics
Some properties of Analytic Functions of
Complex Order
‫بعض خصائص االقترانات التحليلية ذات الرتبة المركبة‬
By
Ala'a Mohammd Ibrahem Al-Shdifat
Student Number
0820401003
Supervisor
Dr. Basem Aref Frasin
2011
‫‪2‬‬
‫بسم هللا الرحمن الرحيم‬
‫جامعة آل البيت‬
‫كلية العلوم‬
‫قسم الرياضيات‬
‫بعض خصائص االقترانات التحليلة ذات الرتبة المركبة‬
‫إعداد الطالبة‪:‬‬
‫االء محمد ابراهيم الشديفات‬
‫إشراف الدكتور‪ :‬باسم عارف فراسين‬
‫التوقيع‬
‫أعضاء لجنة المناقشة‪:‬‬
‫‪ .1‬الدكتور ‪:‬باسم عارف فراسين‬
‫(رئيسا ً ومشرفاً)‬
‫‪………….….‬‬
‫‪ .2‬الدكتور احمد بطاينه‬
‫(عضواً)‬
‫‪………….….‬‬
‫‪ .3‬الدكتور ليث عازر‬
‫(عضوا ً)‬
‫‪…………….‬‬
‫‪ .4‬الدكتو وصفي شطناوي‬
‫(عضواً)‬
‫‪……………..‬‬
‫قدمت هذه الرسالة استكماالً لمتطلبات الحصول على درجة الماجستير في الرياضيات في كلية العلوم‬
‫في جامعة آل البيت‪.‬‬
3
‫بسم هللا الرحمن الرحيم‬
Al al-Bayt University
Faculty of Science
Department of Mathematics
Some properties of Analytic Functions of
Complex Order
By
Ala'a Mohammd Ibrahem Al-Shdifat
Student Number
0820401003
Supervisor
Dr. Basem Aref Frasin
Thesis Defense Committee:
1. Dr. Basem Aref Frasin
Signature
(Supervisor and Chairman)
2. Dr. Ahmad Bataineh
(Member)
3. Dr .Laith Azar
(Member)
4.dr. Wasfi Ahmed Shatanawi
(Member)
...………….….….
….………….…….
…………..……….
………….………..
Thesis submitted in partial fulfillment of the requirements for the degree of Master
Mathematics at Al al-Bayt University.
4
Dedication
To my father and mother who brought me virtue and
from them I learned
the principles
of modesty, love,
tolerance and kindness to all people.
I am especially grateful to express appreciation to my
husband 'Maan" who always beside me, for his unconditional
love and immense patience, support and helping me to
finish this work.
To my brothers and sisters.
To my baby ''mota'z".
To my professors and friends.
Ala'a Al-Shdifat
5
Acknowledgments
First of all, I thank God for giving me the strength, patience and courage to complete this
work to the best of my ability.
I appreciate the support of many people who have contributed to this work. I would
like sincerely thank my supervisor, Dr. Basem Aref Frasin for his help and support. His
comments and suggestion were very helpful in completing this work . His trust in me and
my ability as a graduate student is greatly appreciated. A great dept of gratitude to
commission discussion.
I would like also to acknowledge all the faculty members in Mathematics department
at Al-Albayt university for their support and inspiration and to thesis defense committee:
Dr. Ahmad Bataineh , Dr .Laith Azar and Dr. Wasfi Ahmed Shatanawi .
Finally, I owe special gratitude to my father who inspired me with his lifelong quest
for knowledge and extended family for the unconditional support and for their
encouragement and patience during my study.
6
List of Symbols
Symbol
Meaning
U
Open unit disc.
D
Simply connected domain.
Complex plane.
A
Class of analytic functions.
T
Class of univalent functions with negative coefficients.
S
Class of analytic and univalent functions.
K
Class of convex functions.
K  
S*
S *  
C
C 
Class of convex functions of order  .
Class of starlike functions.
Class of starlike functions of order  .
Class of close-to-convex functions.
Class of close-to-convex functions of order  .
Subordination.
S  (b )
starlike functions of complex order b and type  .
C (b )
convex functions of complex order b and type  .
H
H   
f z   g z 
Class of spiral-like-functions.
Class of spiral-like-functions of order  .
Hadamard product of two analytic functions.
7
Abstract
In this thesis, we study some properties of functions in the class
Q T (, ; , b ) of analytic functions of complex order b and type  defined in the open
unit disk U  z : z  1 . Furthermore, we shall obtain the order of close-toconvexity and some sufficient conditions for two integral operators to be in certain
classes of analytic functions of complex order.
‫‪8‬‬
‫المـلــخــص‬
‫في هذه الرساله نقوم بدراسة بعض خصائص االقترانات في الفئه ) ‪ Q T (, ; , b‬من االقترانات‬
‫التحليله ذات الرتبه المركبه ‪ b‬والنوع ‪ ‬المعرفه على دائرة الوحده ‪. U  z : z  1‬‬
‫باالضافه الى ذلك سنجد رتبة القريب من التحدب وايجاد بعض الشروط الكافيه لمؤثرين تكامليين لكي‬
‫تنتمي هذه المؤثرات لبعض الفئات من االقترانات التحليله ذات الرتبه المركبه‪.‬‬
9
Table of Contents
Subject
Page
Dedication
I
Acknowledgments
II
List of Symbols
III
Abstract
IV
Abstract(in Arabic)
V
Table of Contents
VI
Introduction
VII
Preliminaries
Chapter
One
1.1 Introduction.
1
1.2 Convex and starlike functions.
6
1.3 Close –to-convex and spirallike functions.
10
1.4 Hadamard product of analytic functions.
13
1.5 Univalent functions with negative coefficients.
15
1.6 Subordination principal.
16
Family of analytic functions of complex order
Chapter
Two
Chapter
Three
1
18
2.1 Introduction.
18
2.2 Coefficient inequalities
21
2.3 Growth and distortion theorems.
24
2.4 Radii of close-to-convexity, starlikeness and convexity
26
2.5 Integral means inequalities.
28
2.6 Neighborhoods of the class QT (, ; , b ) .
30
Order of close-to-convexity and integral operators of analytic
functions of complex order
32
3.1 Introduction.
32
3.2 Order of close-to-convexity
34
3.3 Some sufficient conditions
37
References
41
10
Introduction

Given two analytic functions ƒ and g where   z   z   an z
n
n 2
and g  z   z  b n z n , their Hadamard product or convolution ƒ  z   g  z  is

n 2
defined by

ƒ  z   g  z   z   an b n z .
n
n 2
Making use of the above product, Frasin [11] introduced and studied the class
Q T (, ; , b ) of analytic functions of complex order b and type  defined by
 1  f ( z ) ( z )  
Re 1  
 1    .
 b  f ( z )  ( z ) 
This thesis consists of three chapters.
In chapter one, we shall state only those concepts and results of the theory of univalent
functions and related topics that we shall need in the subsequent chapters of this thesis.
In chapter two, we study the class Q T (, ; , b ) of analytic functions of complex
order b and type  introduced by Frasin [11]. Coefficient inequalities, distortion theorems,
closure theorems, radii of close-to-convexity, starlikeness, convexity for functions
belonging to the class Q T (, ; , b ) are determined. Furthermore, we obtained the
integral means inequality for the function f  z  belongs to the class Q T (, ; , b ) with
the extremal function of this class. Also, we considered p   -neighborhood for functions
in this class.
11
Finally, in chapter three, the order of close-to-convexity for functions in the class K ( ,  )
obtained by Frasin [9] will be given. Furthermore, we proved some sufficient conditions
z
for the integral operators and F1 ,..., n (z )   (f 1 '(t ))1 ...(f n '(t )) n dt
to be in the classes
0
S* (b ) and C (b ) of analytic functions of complex order
introduced by Frasin [9].
b
and type  (0    
12
Chapter I
Preliminaries
In this chapter , we shall state only those concepts and results of the theory of
univalent functions and related topics that we shall need in the subsequent chapters of this
thesis. These results have been given in the form of definitions, lemmas or theorems.
Detailed proofs and further discussions on the subject may be found in the standard texts
of Nehari [27], Goluzin [14], Pommerenke [36], Duren[7], and Goodman [15].
1.1 Introduction
Definition 1.1.1 A single-valued complex function ƒ is said to be univalent in a simply
connected domain D , if ƒ  z
1
  ƒ  z  then z
2
1
 z 2 , z 1 , z 2 D .
Several neccssary and sufficient conditions have been obtained for a function ƒ to be
univalent in D , for example:
(I)
If ƒ  z  is univalent in D then ƒ  z   0 in D .
(II)
If D is a convex domain and if Re  ƒ  z    0 throughout D , then ƒ  z  is
univalent in D .
Note that there exist functions which are univalent but not analytic in a given
domain D . For example, the function
ƒ z  
a z b
c z d
a d  b c  0,
(1.1.1)
is univalent in D , but it has a simple pole at z  d c if it is one of the values taken by
z in D .
On the other hand, an analytic function need not be univalent. For example, ƒ  z   e is
z
analytic in any domain D but is not univalent in D since ƒ  z  2n  i   ƒ  z
an integer.
 and n is
13
In the study of univalent analytic function, the Riemann mapping theorem plays a
crucial role. The theorem states that if D is a simply connected domain and z 0 is a given
point in D then there exists a unique univalent analytic function ƒ which maps D
conformally onto the open unit disk U  z : z  1 and has the properties   z
0
  0,
   z 0   1 , as shown in Figure 1.1
w 0  ƒ  z 0   ƒ  z 0   1  0
y
v
z0z0
1
-1
x
u
W-plane
z-plane
p
Figure 1.1 The Riemann Mapping Theorem
Thus in view of Riemann mapping theorem, instead of studying the properties of
univalent analytic functions in an arbitrary simply connected domain, we shall confine
ourselves to the case in which D is the open unit disk U .
We shall denote by A the class of all those functions ƒ which are analytic in U
and normalized by the conditions f (0)  0 , and ƒ  0   1 . Thus each ƒ  A has a
Taylor series expansion of the form

ƒ  z   z   an z
n
.
(1.1.2)
n 2
Further, by S we shall denote the class of all functions ƒ  A which are
univalent in U . The leading example of a function of the class S is the Koebe function

k ( z )  z 1  z   z   n z
2
n 2
n
,
(1.1.3)
14
which maps the unit disk U onto the entire plane minus the part of the negative real axis
from
1
to - , as shown in Figure 1.2.
4
Other simple examples of functions in S are:
(i)
ƒ  z   z , the identity mapping;
(ii)
ƒ  z   z 1  z  , which maps U conformally onto the half plane Rew  
1
, as shown in Figure 1.2
w  k (z )  z / 1  z 
2
y
v
1
-1
x
u
-1/4
W-plane
z- Plane
Figure 1.2 Koebe Function
w  I ( z )  z / 1  z 
y
v
1
-1
x
u
-1/2
z- plane
w-plane
Figure 1.3 the Identity Function
1
2
15
It is easy to check that the sum, difference, product and quotient of two functions
in S need not be in S . The same is true for the derivative and integral of a function ƒ  S .
However, the composition of two functions in S , and the inverse of a function in S are
in S . The class S is also preserved under a number of elementary transformations.
Here is a partial list:
(i)
Conjugation: if ƒ  S and g  z   ƒ  z

  z   a z , then g  S .
n
n
n 2
(ii)
Rotation: if ƒ  S and g  z   e  i  ƒ(e i  z ) , then g  S .
(iii) Dilation: if ƒ  S and g  z   r ƒ  rz  , where 0  r  1 , then g  S .
1
Koebe [19] proved that there is a constant k (Koebe's constant) such that the
boundary of the image of U under w    z   S is always at a distance not less than
k from w  0 . This discovery attracted many other mathematicians, including
Bieberbach [1], Gronwall [13], and Pick [33], to mention a few, to the field of univalent
functions.
Bieberbach[1] proved that k 
1
. In fact he proved the following results:
4
Theorem 1.1.1: (Koebe covering theorem). If ƒ  S , then
w :
w  1
4
  ƒ U  .
Theorem 1.1.2: (Growth theorem). For each ƒ  S ,
r
1  r 
2
  z  
r
1  r 
2
,
z
 r  1 .
(1.1.4)
Theorem 1.1.3: (Distortion theorem). For each ƒ  S ,
1 r
1  r 
3
 ƒ  z  
1 r
1  r 
3
,
z
 r  1 .
(1.1.5)
16
Theorem 1.1.4 For each ƒ  S ,
1 r
1 r

zf (z )
f (z )

1 r
1 r
,
( z  r  1).
(1.1.6)
The results in Theorems 1.1.1, 1.1.2 and 1.1.3 are sharp for the Koebe function.
Bieberbach [1] also proved that a2  2 ,  ƒ  S . The Koebe function was acting
as an extremal function for most of the results proved and since a  n , n  2 , he
n
made the following conjecture:
Bieberbach conjecture:

If ƒ  z   z   an z  S then a  n ,  n  2 .
n
n
n 2
This famous problem stood as a challenge until 1985 and was settled by De Branges [13]
by giving a surprisingly short proof in view of the rich and continuing history of the
problem.
However, the Bieberbach conjecture has inspired the development of many
important and ingenious methods which now form the backbone of the subject. In fact,
several subclasses of S have been introduced by a fixed geometric property of the image
domain. It is interesting to note that these subclasses play an important role in other
branches of mathematics.
For example, starlike function play an important role in the solution of certain
differential equations ((Gabriel, [12]), (Robertson,[37]), (Saitoh, [45]), (Owa et al.,[30]))
whereas spirallike functions have been the source of useful counter examples in geometric
function theory ((Epstien and Schoenberg, [8]), (Krzyz and Lewandowski, [19])). Some
of these classes are as follows:
17
1.2 Convex and starlike functions.
Definition 1.2.1 A function ƒ  S is said to be convex in U if the image of U under ƒ
is a convex region i.e., the line segment joining any two points of ƒ U
 lies entirely
in ƒ U  , as shown in Figure 2.1.
Let K denotes the class of all functions of S which are convex in U .
w   z  K
y
v
1
-1
x
u
w-plane
z- plane
Figure 2.1 Convex Function
A necessary and sufficient condition for a function ƒ  S to be convex has been
given by Robertson [37].
Theorem 1.2.1 A function ƒ  K if and only if

z   z  

  z  
Re 1 
  0, z U .
Robertson [36] also introduced the concept of order for functions in K .
(1.2.1)
18
Definition 1.2.2 A function ƒ  K is said to be convex of order  (0    1) if and only
if

Re 1 
z   z  

   , z U .
(1.2.2)
  z  
We shall denote this class by K   . It is clear that K  0   K .
Some of the results for convex function are as follows:

Theorem 1.2.2[30] If ƒ  z   z   an z n K , then a  1 for n  2, 3,
n
..
n 2
This theorem is sharp with the extremal functions of the form   z   z 1  xz

1
where x  1 .
Theorem 1.2.3 [3] If ƒ  z   K   , then
Re
 
f (z )
z

2
2 1
2  1
,
1
for (0    ) .
2
A subclass of S which is wider than K is the class of starlike functions.
Definition 1.2.3 A function ƒ  S is said to be starlike in U if the image of U under ƒ
containing the point w  0 is starlike region i.e., the line segment joining w  0 to any
point of ƒ U
 lies entirely in ƒ U  , as shown in Figure 2.2.

We shall denote by S the class of all functions of S which are starlike with
respect to the origin.
19
w  ƒz   S 
y
v
1
-1
x
u
z- Plane
W-plane
Figure 2.2 Starlike Function
A necessary and sufficient condition for ƒ  S to be starlike is also due to
Robertson [37].
Theorem 1.2.4 A function ƒ  S if and only if

 z ƒ  z  
  0, z U .
ƒ
z




Re 
(1.2.3)
As in the case of convex functions, the order  for starlike functions has been defined by
Robertson [37].
Definition 1.2.4 A function ƒ  S is said to be starlike of order  (0    1) if and

only if
 z ƒ  z  

 ƒ z  
Re 
z U .
(1.2.4)
20
We shall denote by S    , the class of starlike functions of order  (0    1) in U .
Clearly S   0   S  .
It is clear from Definition 1.2.1 and 1.2.3 that K  S . Also from (1.2.2) and (1.2.4) we

have

ƒ  K   iff z ƒ  z   S   ,
ƒ  S   iff

z

0
ƒ t 
(1.2.5)
dt  F  z   K   .
t
Some of the results for starlike functions are as follows:

Theorem 1.2.5 [38] If ƒ  z   z   an z  S    then
n
n 2
n
an 
  k  2 
k 2
 n  2 , 3,  .
,
 n  1!


Theorem 1.2.6 [37]: If ƒ  z   z   an z  S   , then
n
n 2
r
1  r 
21 
  z


r
1  r  
2 1 
z
,
 r  1 .
The results in Theorem 1.2.3 and 1.2.4 are sharp with the extremal function
S

z  
z
1  z 
2 1  
.
(1.2.6)
21
1.3 Close  to  convex and spirallike functions.
Another interesting subclass of S which contains S

is the class of close-to-convex
functions, introduced by Kaplan [18].
Definition 1.3.1 A function ƒ  A is said to be close-to-convex in U if there is a
convex function  such that
 ƒ  z  
0


z




Re 
z U ,
(1.3.1)
we denote this class of close-to-convex functions by C .
Remark 1.3.1 The condition (1.3.1) is equivalent to
 z ƒ  z  
0
g
z




Re 
z U ,
for some g  S and z U .

Kaplan [18] proved that every close-to-convex function is univalent; thus C  S . Every
convex function is obviously close-to-convex. More generally, every starlike function is
close-to-convex. These remarks are summarized by the chain of proper inclusions
KS

 C S .
The geometric interpretation of close-to-convex functions is given by Kaplan[18]. He
further characterized close-to-convex functions, without reference to the convex function
 in following way:
22
Theorem 1.3.1 Let ƒ be analytic and ƒ  z   0 in U , then ƒ 
z   z  

2
 Re 1 
d    ,
  z  

1
if and only if
where    , z  re and r  1 .
i
1
2
The order  for ƒ  C has been introduced by Libera [21].
Definition 1.3.2 A function ƒ  A is said to be close-to-convex of order  , ƒ  C  , if
  K such that
   z  
 ,


z




Re 
(1.3.2)
for some  (0    1) and z U . Clearly C 0   C .
Remark 1.3.2 The condition (1.3.2) is equivalent to
 z ƒ  z  
 ,
g
z




Re 

for some g  S ,  (0    1) and z U .
A class wider than the class of starlike function is the class of spirallike function
introduced and studied by Spacek [45].
Definition 1.3.3 A function ƒ  A is said to be spirallike in U if and only if
 z ƒ  z  
0
 ƒ z  
Re 
for some  with   1 .
z U ,
23
It was shown by Spacek [45] that spirallike functions are univalent in U.
Taking   e i  ,  

2
, Libera [21] introduced the class H  of   spirallike function.
Definition 1.3.4 A function ƒ  A is said to be a
  spirallike function; ƒ H  , if
and only if


Re e
for some real 

i
z ƒ  z
ƒz

  0,
 

  2 and z U .
Libera[21] introduced and studied the notion of order  (0    1) for some
  spirallike function in U .
Definition 1.3.5 A function ƒ  A is said to be a   spirallike function of order  if
and only if

Re e
for some real 

i
z ƒ  z 
ƒ z 

z U ,
  cos 
   2  and for some  (0    1) .
The class of A -spirallike function of order a, we shall denote by H   .

Clearly H    S   , H
0

0
 0  S

and H

 0  H .

24
1.4 Hadamard product of univalent functions.
Now we define the Hadamard product of two analytic functions which will be used
throughout the thesis.
Definition 1.4.1 [7]: Given two functions ƒ, g  A , where

  z   z   an z

n
and g  z   z   b n z ,
n
n 2
n 2
their Hadamard product or convolution ƒ  z   g  z
 is defined by

ƒ  z   g  z   z   an b n z
n
z U .
(1.4.1)
n 2
This product is associative, commutative, and distributive over addition. Note that
the function

I  z   z 1  z   z   z
1
n
acts as the identity element under
n 2
convolution.
Pólya and Schoenberg[35] conjectured that if ƒ  K and g K , then
ƒ  g  K . This problem has a long and illustrious history. Many mathematicians
attacked this problem and obtained partial results. The first major result towards verifying
the conjecture was due to Suffridge [50]. He showed that Hadamard product of two
convex functions is close-to-convex, hence univalent. The conjecture was finally proved
by Ruscheweyh and Sheil-Small [44]. In fact, they proved the following results:
25
Theorem 1.4.1 If ƒ and g  K , then ƒ  g  K .
Theorem 1.4.2 If ƒ K and g C , then ƒ  g C .
An equivalent formulation of Theorem 1.4.1 is:
Theorem 1.4.3 If ƒ and g  S , then ƒ  g  S .



Theorem 1.4.4 If ƒ  K , g  S , and

is analytic function with positive real part in U ,
then  ƒ   g  ƒ  g  is also analytic function with positive real part.
Recently, Silverman et al. [46] gave characterization for convex and starlike
functions in terms of convolutions. They proved the following results.
Theorem 1.4.5 ƒ  K   , 0    1 if and only if
1 
ƒ z  
z 
2
z   x    1    z 
1  z 
3
  0,



z  r  1, x  1 .
Theorem 1.4.6 ƒ  S   , 0    1 if and only if

1 
  z  
z 
z  (x  2  1)  2  2  z 
2
1  z 
2
  0,


z  r  1, x  1 .
26
1.5 Univalent functions with negative coefficients.
Definition 1.5.1 Let T denote the subclass of S consisting of functions whose
nonzero coefficients, from the second coefficient onwards, are negative. That is, an
analytic and univalent function ƒ is in T if it can be expressed as

ƒ  z   z   an z .
n
(1.5.1)
n 2
Further,
Let S    S    T , K    K    T , and P    P    T .


Clearly the functions in the classes S   and K   are respectively, starlike of

order  and convex of order  with negative coefficients.
Silverman [47] obtained the following theorems for the classes S    and K   .


Theorem 1.5.1 Let f (z )  z   an z . If
 (n   ) | a
n
n 2
n
|  1   , then ƒ  S    .
n 2

Theorem 1.5.2 Let f (z )  z   an z . If
n

 n (n   ) | a
n 2
n
|  1   , then ƒ  K   .
n 2
In 1975 Silverman proved the following theorems for functions in the classes T ,
S   and K   .

Theorem 1.5.3 A function ƒ  T if and only if

n
an  1 .
(1.5.2)
n 2
Theorem 1.5.4 A function ƒ  S    if and only if
n 
  1   

an  1 .
(1.5.3)
n 2
Theorem 1.5.5 A function ƒ  K   if and only if


n 2
n n   
1
an  1.
(1.5.4)
27
1.6 Subordination Principal.
Definition 1.6.1 For two functions f and g analytic in U, we say that the function
f (z ) is subordinate to g (z ) and we write f
g or f (z )
g (z ) ;  z U  , if there
exists a Schwarz function w ( z ) analytic in U with w(0) = 0 and w (z )  1 ,  z U  ,
such that f (z )  g (w (z )) ,  z U  .
The concept of subordination between analytic functions can be traced back to
Lindelof [22], although Littlewood [23,24] and Rogosinski [39,40] introduced the term
and established the basic results involving subordination.
As an example, if n is a positive integer, then z n
z and z 2 n
z 2 in U , but
z 2 n 1 is not subordinate to z 2 in U .
Making use of this concept Janowski [34], introduced the following
important classes of analytic functions.
Definition 1.6.2 For 1  B  A  1 , a function p (z ) analytic in U with p (0)  1 ,
is said to belong to the class P (A , B ) if
p (z )
1  Az
.
1  Bz
(1.6.1)
Definition 1.6.3 A function f (z )  A is said to be in the class S  (A , B ) if
z ƒ  z 
ƒ z 
 P (A , B ) .
(1.6.2)
28
Definition 1.6.4 A function f (z )  A is said to be in the class K (A , B ) if
1
z ƒ ''  z 
ƒ 'z 
 P (A , B ) .
We note that S  (1, 1) = S  and K (1, 1) = K .
(1.6.3)
29
Chapter II
Family of analytic functions of complex order
2.1 Introduction
Very recently, Frasin [11] introduced the following classes of analytic functions of
complex order b (b 
 {0}) and type  (0     defined as follows:
Definition 2.1.1 A function f (z )  A is said to be starlike of complex order
b (b 
 {0}) and type  (0     , that is f (z )  S (b ) if and only if
 1  zf '(z )  
Re 1  
 1   

 b  f (z )
(z U ;b   {0}).
(2.1.1)
Definition 2.1.2 A function f (z )  A said to be convex of complex order
b (b 
 {0}) and type  (0     , denoted by C (b ) if and only if
 1 zf (z ) 
Re 1 
 
 b f (z ) 
(z U ; b   {0}).
(2.1.2)
Definition 2.1.3 A function f (z )  A is said to be the in the class P b  if and only if
 1

Re 1   f (z )  1   
 b

(z U ; b 
 {0}).
(2.1.3)
30
Remark 2.1.1: We note that S0* (b )  S * (b ) and C0 (b )  C(b ) the classes considered
earlier by Nasr and Aouf [ 26] and Wiatrowski [51 ] . Also we note that
S* (e i  cos  )  H    , (  

2
, 0    1) , the class of ,  -spirallike functions of order
 , was introduced by Libera [21] and C=  (e i  cos  )  C , (  

2
, 0    1) , the
class of ¸  -Roberton function of order  was introduced by Chichra [5].
Let T denoted the subclass of A whose member have the form:

  z   z   an z
n
(a n  0) ,
(2.1.4)
n 2
we denote by S* [b ], C [b ] and P [b ] respectively, the classes obtained by taking the
intersections of S* b  , C b  and P (b ) with T ,that is,
S* [b ]  S* b  T , C [b ]  C b  T and P [b ]  P (b ) T .
We obtain that the above classes by using the following class defined by Frasin [9].
Definition 2.1.4 Given b (b 
 {0}) and  (0     . Let the functions


n 2
n 2
(z )  z    n z n and  (z )  z    n z n ,
be analytic in U, such that  n  0, n  0 and  n  n for n  2 , we say that
f (z )  A is in Q T (, ;  , b ) if f (z )   (z )  0 and
 1  f (z ) (z )  
Re 1  
 1   
 b  f (z )   (z )  
(z  U) .
31
Further, let
Q T (, ; , b )  Q (, ;  , b ) T .
We note that, by suitably choosing  (z ),  (z ) we obtain the above subclasses of T of
complex order b and type 
 z

 z  z2

z
z
*
QT 
,
;

,
b

S
b
;
Q
,
;  , b   C b  ;

 
T 
2
3
2
 1  z  1  z

 1  z  (1  z )





and
 z

QT 
, z ; , b   P b .
2
 (1  z )

In fact many new subclasses of T of complex order b and type  can be defined
and studied by suitably choosing  (z ),  (z ) .
For example

 1  f (z )  
 z
 
QT 
, z ; , b   f (z )  T : Re 1  
 1  >   ,
 1 z
 

 b z

and
 z z 2
 

 1

QT 
, z ; , b   f (z )  T : Re 1   (zf '(z )) ' 1 >   ,
3
 b


 (1  z )
 
and so on.
In this chapter, we shall obtain coefficient inequalities, distortion theorems, closure
theorems, radii of close-to-convexity, starlikeness, convexity for functions belonging to
the class Q T (, ; , b ) . Furthermore, we obtain the integral means inequality for the
function f  z  belongs to the class Q T (, ; , b ) with the extremal function of this
class. Also, we consider p   -neighborhood for functions in this class.
32
2.2 Coefficient inequalities
Theorem 2.2.1 Let the function f (z ) defined by (2.1.4) be in the class Q T (, ; , b ).
Then

 Re(b )
n 2
n
 ((1   ) b  Re(b )) n  an  b (1   ).

2
2
(2.2.1)
The result (2.2.1) is sharp.
Proof. Suppose that f  z  QT (, ; , b ) . Then
 1  f (z ) (z )  
Re 1  
 1   
b
f
(
z
)


(
z
)



z U ,
or, equivalently
  
n 1  
 1   (n  n )an z  
    1
Re   n  2 
b  1   n an z n 1  

 
n 2

z U .
Now choose values of z on the real axis and let z  1 through real values to find that
 
  (n   n ) an
 n 2 
 1  a

n
n


n 2
where

 1
 Re    1 ,
 b


33
 
  (n  n ) an
 n 2 
 1  a

n
n


n 2

 Re(b )

 1 ,
2
 b


and so

 (
n 2
n
 n ) an 
2



(1   ) 1   n an 
Re(b )
 n 2
 ,
b
which is equivalent to (2.2.1). The equality in (2.2.1) holds true for the functions
f (z ) defined by
b (1   )
2
f (z )  z 
(Re(b ))n  ((1   ) b  Re(b )) n
2
n 2 .
zn
(2.2.2)
Corollary 2.2.1 Let the function f (z ) defined by (2.1.4) be in the class Q T (, ; , b )
then
b (1   )
2
an 
n 2
(Re(b ))n  ((1   ) b  Re(b )) n
2
.
(2.2.3)
The result (2.2.3) is sharp for the function f (z ) given by (2.2.2)
Putting (z )  z  (1  z )2 and  (z )  z / (1  z ) in Theorem 2.2.1, we have
Corollary 2.2.2 Let the function f (z ) defined by (2.1.4) be in the class S * [b ] .Then

 Re(b )n  ((1   ) b
n 2
2
 Re(b ))  an  b (1   ).

2
The result is sharp for
b (1   )
2
f (z )  z 
Re(b )n  ((1   ) b  Re(b ))
2
zn
n  2.
34
Putting (z )  (z  z 2   (1  z )3 and (z )  z / (1  z )2 in Theorem (2.2.1), we have
Corollary 2.2.3 Let the function f (z ) defined by (2.1.4) be in the class C* b  .
Then

 Re(b )n
n 2
2
 ((1   ) b  Re(b ))n  an  b (1   ) .

2
2
The result is sharp for
b (1   )
2
f (z )  z 
Re(b )n  ((1   ) b  Re(b ))n
2
2
zn
n 2
Putting (z )  z  (1  z )2 and  (z )  z in Theorem 2.2.1, we have
Corollary 2.2.4 Let the function f (z ) defined by (2.1.4) be in the class p* b 

 (Re(b ))n  a
n
n 2
 b (1   ).
2
The result is sharp for
b (1   )
2
f (z )  z 
(Re(b ))n
n  2.
zn
For the notational convenience we shall henceforth denote.
 n ( , b )  (Re(b ))n  ((1   ) b  Re(b )) n
2
n  2.
35
2.3 Growth and distortion theorems
Theorem 2.3.1 Let the function f (z ) defined by (2.1.4) be in the class QT (, ; , b). If
 n ( , b )n 2

is a non-decreasing sequence, then
b (1   )
2
z 
b (1   )
2
z
 2 ( , b )
2
 f (z )  z 
 2 ( , b )
2
z
,
(2.3.1)
where  2 ( , b )  (Re(b ))2  ((1   ) b  Re(b )) 2 . The equality in (2.3.1) is attained
2
for the function f (z ) given by
b (1   )
2
f (z )  z 
 2 ( , b )
z2
.
(2.3.2)
Proof. Note that


n 2
n 2
 2 ( , b ) an    n ( , b ) an  b (1   ) ,
2
or, equivalently


n 2
b (1   )
2
an 
 2 ( , b )
(2.3.3)
,
this last inequality follows from Theorem (2.2.1). Thus we have

f (z )  z   an z
n
 z  z
n 2
2
b (1   )
2


an  z 
n 2
 2 ( , b )
z
2
,
(2.3.4)
and

f (z )  z   an z
n 2
n
 z  z
2


n 2
b (1   )
2
an  z 
 2 ( , b )
z
2
(2.3.5)
,
36
for z U . From the inequalities (2.3.4) and (2.3.5) we obtain the inequality (2.3.1).
Theorem 2.3.2 The disk |z| < 1 is mapped onto a domain that contains the disk
 ( , b )  b (1   )
 n
,
 2 ( , b )
2
by any f (z ) Q T (, ;  , b ) . The theorem is sharp with the function f (z ) given by
(2.3.2).
Theorem 2.3.3 Let the function f ( z ) defined by (2.1.4) be in the class Q T (, ; , b ) .
If  n ( , b ) / n n  2 is a non-decreasing sequence, then

2 b (1   )
2
1
2 b (1   )
2
z  f '(z )  1 
 2 ( , b )
 2 ( , b )
z .
(2.3.6)
The equality in (2.3.6) is attained for the function f ( z ) given by (2.3.2).
Proof. In view of Theorem 2.2.1
 2 ( , b )
2

n
n 2

an    n ( , b ) an  b (1   ) ,
2
(2.3.7)
n 2
that is,

n
n 2
2 b (1   )
2
an 
Form (2.3.8), we can easily prove that
 n ( , b )
.
(2.3.8)
37

f '(z )  1   n an z
n 1
 1 z
n 2

n
n 2
2 b (1   )
2
an  1 
 2 ( , b )
z ,
(2.3.9)
and

f '(z )  1   n an z
n 1
 1 z
n 2

n
n 2
2 b (1   )
2
an  1 
 2 ( , b )
z
,
(2.310)
for z U . Combining the inequalities (2.3.9) and (2.3.10) we obtain the inequality
(2.3.6).
2.4 Radii of close-to-convexity, starlikeness and convexity
Theorem 2.4.1 Let the function f ( z ) be defined by (2.1.4) be in the class
QT ( , ; , b). Then f ( z ) is close-to-convex of complex order b in z  r1 , where
1/ ( n 1)
  ( , b ) 
r1  r1 ( , b )  inf  n

n
 b n (1   ) 
(n  2) .
(2.4.1)
The result is sharp for the function f ( z ) being given by (2.3.2).
Proof. We must show that f '(z ) 1  b for z  r1 , where r1 is given by (2.4.1). From
(2.1.4) we have

f '(z )  1   nan z
n 1
.
n 2
Thus f '(z ) 1  b if
 n
n 2  b

 

 an z

but, by Theorem 2.2.1, (2.4.2) will be true if
n 1
1 ,
38
 n 
  z
 b 
n 1

 n ( , b )
2
b (1   )
,
(2.4.2)
(n  2) .
(2.4.3)
that is, if
1/ ( n 1)
  ( , b ) 
z  n

 b n (1   ) 
Theorem 2.4.1 follows easily from (2.4.3).
Theorem 2.4.2 Let the function f (z ) be defined by (2.1.4) be in the class Q T ( , ; , b ) .
Then f ( z ) is starlike of complex order b in z  r2 , where
1/ ( n 1)


 n ( , b )
r2  r2 ( , b )  inf 

n
 b (n  b  1)(1   ) 
n2
The result is sharp for the function f (z) being given by (2.3.2).
Proof. It is sufficient to show that
zf (z )
1  b ,
f (z )
for |z| < r2, where r2 is given by (2.4.4). From (2.1.4) we find that

zf (z )
1 
f (z )
 (n  1)a
n 2
n

1   an z
n 2
z
n 1
n 1
.
.
(2.4.4)
39
Thus
zf (z )
 1  b if
f (z )
 n  b 1 
 an z
b
n 2 


 
n 1
1 ,
(2.4.5)
but, by Theorem 2.2.1, (2.4.5) will be true if
 n  b 1 

 z
b


n 1

 n ( , b )
,
2
b (1   )
that is, if
1/ ( n 1)


 n ( , b )
z 

 b (n  b  1)(1   ) 
(n  2) .
(2.4.6)
Theorem 2.4.2 follows easily from (2.4.6).
Corollary 2.4.1 Let the function f ( z ) be defined by (2.1.4) be in the
class Q T ( , ; , b ) . Then f ( z ) is convex of complex order b in z  r3 , where
1/ ( n 1)


 n ( , b )
r3  r3 ( , b )  inf 

n
 b n (n  b  1)(1   ) 
(n  2) .
The result is sharp for the function f ( z ) being given by (2.3.2).
2.5 Integral Means Inequalities
The following subordination result will be required in our present investigation.
Lemma 2.5.1 [24]: If f and g are analytic in U with f
2

0

2

g then
f (re ) d    g (re i  ) d 
i
0
,
40
where   0 , z  re i  and 0  r  1 .
Applying Theorem 2.2.1 and Lemma 2.6.1, we prove the following
Theorem 2.5.1 Let   0 . If f (z ) Q T (, ;  , b ) and  n ( , b )n  2 non-decreasing

sequence, then, for z  re i  and 0  r  1 we have
2
2



f (re ) d    f 2 (re i  ) d 
0
0
i
,
where f 2 (z )  z  b (1   ) /  2 ( , b )z 2 .
2
Proof. Let

f (z )  z   an z n
(an  0, z U )
n 2
and
f 2 (z )  z  b (1   ) /  2 ( , b )z 2 ,
2
then we must show that
2

0
n 2
 1 a z

2
n
b (1   )

2
d   1
n 1
0
 2 ( , b )
z d.
By Lemma 2.5.1, it suffices to show that

1   an z
b (1   )
2
n 1
1
n 2
 2 ( , b )
z.
Setting

1   an z
n 2
from (2.5.2) and (2.2.1), we obtain
b (1   )
2
n 1
 1
 2 ( , b )
 (z ) ,
(2.5.1)
41
 (z ) 


n 2
 z


n 2
 2 ( , b )
an z n 1
2
b (1   )
 2 ( , b )
an  z .
2
b (1   )
This completes the proof of the theorem.
Remark 2.5.1 Taking different choices of  ( z ) and  (z ) in Theorem 2.5.1, we can
obtain integral means inequalities for functions belonging the classes S * [b ] , C* b 
and Р b  .
2.6 Neighborhoods of the class Q T (, ; ,b ) .
For f  T of the form (2.1.4) and   0 , we define




M  (f )   g  T : g (z )  z  b n z n ,  n p 1 an  b n   
n 2
n 2


which was called the p   -neighborhood of f. So, for e(z) = z, we see that




M  (e )   g  T : g (z )  z  b n z n ,  n p 1 b n   
n 2
n 2


where p is a fixed positive integer. Note that M o (f )  N  (f ) and
M 1 ( f )  M  ( f ). N ( f ) is called a  neighborhood of f by Ruscheweyh [42] and
M  (f ) was defined by Silverman [47].
In this section, we consider p   -neighborhood for function in the class
42
Q T (, ; , b ) .
Theorem 2.6.1 If  n ( , b ) / n p 1

n 2
is a non-decreasing sequence, then,
Q T (, ;  , b )  M p (e )
,
where   2 p 1 b (1   ) /  2 ( , b )
2
Proof. It follows from (2.2.1) that if f (z ) Q T (, ;  , b ) , then

n
n 2
2 p 1 b (1   )
2
p 1
an 
 2 ( , b )
.
This gives that Q T (, ;  , b )  M p (e )
Putting (z )  z / (1  z )2 and  (z )  z / (1  z ) in Theorem 2.6.1, we have
2
2
Corollary 2.6.1 S* b   M P (e ) , where   2 p 1 b (1   ) / Re(b)  (1   ) b 

.
Putting (z )  (z  z 2 ) / (1  z )3 and (z )  z / (1  z )2 in Theorem 2.6.1, we have
2
2
Corollary 2.6.2 C b   M P (e ) , where   2 p b (1   ) / Re(b)  (1   ) b 


Putting (z )  z / (1  z )2 and  (z )  z in Theorem 2.6.1, we have
Corollary 2.6.3 P b   M P (e ) , where   2 p b (1   ) /Re(b).
2
43
Chapter III
Order of close-to-convexity and integral operators of analytic
functions of complex order
3.1 Introduction
A function f (z )  A is said to be close-to-convex of complex order  (   0) ,
and type   , if there exists a function g (z ) belonging to S ( ) such that
 1 zf   z 

Re 1  (
 1)    ,
  g z 

 z U 
(3.1.1)
we denote by K ( ,  ) the subclass of A consisting of functions which are close-to-convex
of complex order  and type  in U . We note that the class K (1, 0) is the class
of close-to-convex functions introduced by Kaplan [18] and Ozaki [31].
Pfaltzgraff et al.[32] have proved that if f (z ) in A satisfies the condition
 1 zf   z  
Re 1 
 
'
  f (z ) 
1
(    1),
2
then f (z ) in the class S . Furthermore, Cerebiez-Tarabicka et al. [4] have shown that if
f (z ) in A satisfies the condition
 zf   z  
1
Re 1 

f (z ) 
2

1
(    1),
2
then
 zf   z  
Re 
  0
 g z  
 z U 
.
Recently, Owa [29] proved that if f (z ) in A satisfies the condition
44
 zf ''  z  


Re 1  '
0
f (z ) 



 z U 
,
then
 zf   z   3
Re 
 
 g z   5
 z U 
,
where g (z )  S * ( / (  1)),   0.
Also, Frasin and Oros [10] proved that if the function f (z ) in A satisfies the condition
 zf   z 

Re 
0
 f (z )

 z U 
,
 zf   z  
1
Re 
 
 g  z   2  1
 z U 
,
where g (z )  S * and 1    3  
In this chapter, we proved the order of close-to-convexity for functions in the class
K ( ,  ) obtained by Frasin [9].
In order to show our results, we shall need the following lemma due to Obradoviˇc et
al.[28].
Lemma 3.1.1 Let f  S (b ) , b   0 , and let a   0 with   ab  1 Then
  f  z  a 
2ab
Re  
 2
 z  


 z U  .
45
3.2 Order of close-to-convexity
With the aid of Lemma 3.1.1, Frasin [9] proved the following result.
Theorem 3.2.1 If the functions f (z ) and g (z ) are in A and satisfies the conditions

 1  zf   z   

Re 1  
  0

   f (z )  

 z U 
,
(3.2.1)
with   a  1 ,   b / (a  1); a, b   0; a  1 ,and
 zf (z ) 
 a 1 
Im 
0,
  0 or Im 
 b 
 g (z ) 
then f (z ) belongs to the class K ( ,  ) , where
2ab
 a 1 
.
 b 
  1  (2 a 1  1) Re 
Proof. If we define g (z ) by
1
a  1  zg (z ) 
1  zf (z ) 
 1  1  

,
b  g (z )
b  f (z ) 

(3.2.2)
then from the condition (3.2.1) and (3.2.2), we have g (z )  S ( ) , with   b / (a  1) . It
is easy to see that (3.2.2) implies
 g (z ) 
f '(z )  

 z 
a 1
,
or
zf '(z )  g (z ) 


g (z )  z 
Applying Lemma 3.1.1 to g(z), we obtain
a
.
46
a


 a  1  zf '(z )  
 a  1   g (z ) 

Re 1 
 1   Re 1 
 

  1 
b  g (z )
b  z 






a

 g (z ) 

 a 1  
 1  Re 
Re
 
  1
 b  

 z 

 g (z ) a 

 a 1  
 Im 
 Im 
  1
 b  
 z 


a

 g (z ) 

 a 1  
 1  Re 
 Re 
  1
 b  
 z 


 a 1 
 1  (22a  1) Re 

 b 
2ab
 a 1 
 1  (2 a 1  1) Re 

 b 
This completes the proof of Theorem 3.2.1.
Letting a = 1 in Theorem 3.2.1, we have
Corollary 3.2.1: If the function f C (b ) with 0  b  2 , then f  K (b / 2,  ) , where
  1
21b  2
b
Letting b = 1 in Theorem 3.2.1, we have
Corollary 3.2.2: If the functions f (z ) and g (z ) are in A and satisfies the conditions
 zf ''(z ) 
Re 1 
> 0
f '(z ) 

(z U ) ,
with 0 < 2a ,   1/ (a  1); a   0; a  1, and
Im(a  1)  0 or Im(
zf '(z )
)  0,
g (z )
47
then f (z ) belongs to the class K ( ,  ) , where
  1  (2
2a
a 1
 1) Re(a +1).
Letting b = 1 in Corollary 3.2.1 or a = 1 in Corollary 3.2.2, we have
Corollary 3.2.3: Let the functions f (z ) and g (z ) be in A . If
 zf ''(z ) 
Re 1 
> 0
f '(z ) 

(z  U) .
Then
 zf '(z )  1
Re 

 g (z )  2
(z  U) .
Therefore, if f (z ) is convex in U then f (z ) is close-to-convex of order
Letting b  a 1 in Theorem 3.2.1, we have
Corollary 3.2.4: Let the functions f (z ) and g (z ) be in A . If

1  zf ''(z ) 
Re 1 

 > 0
 a  1  f '(z ) 
(z  U) .
Where 0 < a  1/ 2 , then
 zf '(z )  1
Re 
> a
 g (z )  4
.
Letting a = 1/2 in Corollary 3.2.4, we have
Corollary 3.2.5: Let the functions f (z ) and g (z ) be in A if
1
in U.
2
48
 2  zf ''(z )  
Re 1  
 > 0
 3  f '(z )  
(z U ) ,
then
 zf '(z )  1
Re 
> ,
 g (z )  2
That is, f (z ) is close-to-convex of order
(z U ) .
1
in U .
2
3.3 Integral operators of analytic functions of complex order
Definition 3.3.1 Let f i  A and  i  0 for all i  1, 2,..., n. Then the integral operators
Fn , F1 ,...,n : An  A defined as follows:


1
n
 f 1 (t ) 
 f n (t ) 
Fn (z )   
 ... 
 dt
t 
 t 
0
z
i  0 .
(3.3.1)
And
z
F1 ,..., n (z )   (f 1 '(t ))1 ...(f n '(t )) n dt
i  0
0
.
(3.3.2)
In this section, we proved some sufficient conditions for the above integral operators
to be in the classes S* (b ) and C (b ) obtained by Breaz et al. [2].
49
Some sufficient conditions.
Theorem 3.3.1 Let i , i {1, . . . , n} be real numbers with the properties  i  0
for i {1, . . . , n } , and
n
0  1   i  1 ,
i 1
we suppose that the functions f i  S (b ) , for i  {1, . . . , n} and b   {0} .Then we
n
have the integral operator Fn C  (b ) , where   1    i
i 1
Proof. We calculate for Fn the derivatives of the first and second order. From (3.3.1) we
obtain:


1
n
 f 1 (z ) 
 f n (z ) 

Fn (z )  
 ... 

 z 
 z  ,
n
 zf (z )  f i (z ) 
Fn(z )  i  i
 Fn(z ) .
zf
(
z
)
i 1

i

Then we have
 zf (z )  f 1 (z ) 
 zf n(z )  f n (z ) 
Fn(z )
 2  1
  ...   n 
,
Fn(z )
zf 1 (z )
zf n (z )




 f (z ) 1 
 f ( z ) 1 
Fn(z )
 1  1
   ...   n  n
 .
Fn(z )
f
(
z
)
z
f
(
z
)
z 
 1

 n
(3.3.3)
Multiply the relation (3.3.3) with z we obtain:
 f (z ) 
zFn(z ) n
 i  i
 1
Fn(z ) i 1  f i (z ) 
.
(3.3.4)
50
Multiply the relation (3.3.4) with
1
we obtain:
b
 zf (z )  n
 1 zf (z )
 n
1 zFn(z ) 1 n
 i  i
 1    i  1  ( i
 1)   i .
b Fn(z ) b i 1  f i (z )
 i 1  b f i (z )
 i 1
(3.3.5)
The relation (3.3.5) is equivalent to:
1
 1  zf (z )   n
1 zFn(z ) n
   i 1   i
 1     i  1 .
b Fn(z ) i 1  b  f i (z )
  i 1
By taking the real part of the above inequality we obtain:
 1  zf i (z )   n
 1 zFn(z )  n
Re 1 
 1     i  1 .
    i Re 1  
 b Fn(z )  i 1
  i 1
 b  f i (z )
Since, we apply in the above relation f i  S (b ) for i  {1, . . . , n } the inequality (3.1.1)
where   0 and obtain:
n
 1 zFn(z ) 
Re 1 


i


i 1
 b Fn(z ) 
.
n
n
i 1
i 1
Because 0  1    i  1 , we have that Fn  C (b ) , where   1    i .
Corollary 3.3.1 Let 1  0 . If 0  1  1  1 and the function f 1  S (b ) , then the integral
operator F1  CP (b ) ,where P  1  1 .
Proof. . Putting n = 1 in Theorem 3.3.1, we obtain the result.
51
Remark 3.3.1: Putting b  e i  cos(  

2
) in Theorem 3.3.1, we have the
following corollary
Corollary 3.3.2: Let the functions f i  S* (  

2
) for all i  1, 2,..., n .Then the
n
n
i 1
i 1
integral operator Fn  C ( ), where   1    i ,  i  0 and 0  1    i  1 .
Corollary 3.3.3 Let the function f 1  S* (b ) . Then the integral operator F1  C (b ) ,
where   1  1 and 0  1  1  1 .
Proof. Putting n = 1 in Theorem 3.3.1, we obtain the result.
Remark 3.3.2: Putting b  e  i  cos(  

2
) in Theorem 3.3.1, we have the following
corollary
Corollary 3.3.4: Let the functions f i  S* (  

2
) for all i  1, 2,..., n .Then the
n
n
i 1
i 1
integral operator F1 ... n  C ( ), where   1    i and 0  1    i  1 .
52
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