1
Some properties of a subclass of
analytic functions
Presented by
Dr. Wasim Ul-Haq
Department of Mathematics
College of Science in Al-Zulfi, Majmaah University
KSA
2
Presentation Layout
•
•
•
•
Introduction
Basic Conceps
Preliminary Results
Main Results
3
Introduction
Geometric Function Theory is the branch of Complex Analysis
which deals with the geometric properties of analytic functions.
The famous Riemann mapping theorem about the replacement
of an arbitrary domain (of analytic function) with the open unit
disk E   z  C :| z | 1 , is the founding stone of the geometric
function theory. Later, Koebe (1907) and Bieberbach (1916)
studied analytic univalent functions which map E onto the
domain with some nice geometric properties. Such functions and
their generalizations serve a key role in signal theory, constructing
quadrature formulae and moment problems.
4
Functions with bounded turning, that is, functions whose
derivative has positive real part and their generalizations have
very close connection to various classes of analytic univalent
functions. These classes have been considered by many
mathematicians such as Noshiro and Warchawski (1935), Chichra
(1977), Goodman (1983) and Noor (2009).
In this seminar, we define and discuss a certain subclass of
analytic functions related with the functions with bounded
turning. An inclusion result, a radius problem, invariance under
certain integral operators and some other interesting properties for
this class will be discussed.
5
Basic Concepts
The class A (Goodman, vol.1)[2]
Let A be the class of function f :

f ( z )  z   an z n ,
n2
(1)
which are analytic in the open unit disk E   z  C :| z | 1.
The class S of univalent functions[2]
S   f  A : f ( z1 )  f ( z2 ) for z1  z2 
(2)
6
The class P (Caratheodory functions) [2]
Let P be the class of functions p which are analytic
in the open unit disk E with p(0)  1 and Re p( z )  0, for z in E.
The class P
19()
k (Pinchuk,1971)
k 1


Pk   p : p( z)  (  ) p1 ( z)  ( k  1 ) p2 ( z), p1, p2  P, k  2.
4 2
4 2


6
Some related classes to the class P (Noor, 2007)[6]
We define the following.



(i). P( )   p : p( z ) is analytic in E with p(0) 1 and arg p( z )  , 0    1 .
2


Note that P(1)  P, the class of functions with positive real part.
(ii). Let P k ( ) be the class of functions p analytic in E satisfying p(0)  1 and
k 1
k 1
p( z )     p1 ( z )     p2 ( z ),
 4 2
 4 2
(3)
where k  2, 0    1, pi  P( ), i  1, 2. Note that P 2 ( )  P( ).
5
Examples
(i). For z  E , the function

 1 z 
L ( z )  
  P( ).
 1 z 
(ii). For z  E , the function


 k 1  1 z   k 1  1 z 
p ( z )     
    
 ,
 4 2  1 z   4 2 1 z 
belongs to the class P k ( ).
6
Special classes of univalent functions [2]
Starlike functions (Nevanilinna, 1913)


zf ( z )
S*   f  A :
 P, z  E  .
f ( z)


Convex functions (Study, 1913)

( zf
C   f  A:
f

( z ))

 P, z  E  .
( z )

Alexander relation (1915)
f  C if and only if zf   S * .
11
Functions with bounded turning and related classes
R   f  A : f ( z )  P , z  E . ([2, vol.1, p.101])
R ( 1 )   f  A :  f ( z )  1 zf ( z )   P, 1  R, z  E  (Chichra [1]).
It is well-known that functions in the class R are univalent.
J. Krzyz [3] disproved that R  S * .We observe that
R ( 1 )  R (0)  R, for 1 >0.
12
Convolution (or Hadamard Product)
Hadamard product (or convolution) of two analytic functions f given by (1)
n
and g ( z )  z   bn z n , is defined by
n2
f
n
 g  ( z )  z   an bn z n , z  E.
n2
Lemma1 (Singh and Singh)[9]
1
, z  E,
2
then for any function F analytic in E ,the function p  F
takes the values in the convex hull of F ( E ).
If p is analytic in E , p (0)  1 and Re p ( z )>
13
A class of analytic functions [Noor and Haq, 7]


 



R k ( ,  ,  )   f  A : 
f ( z ) 
 zf ( z )    P k ( ), z  E  ,
 
  



where  ,   0,     0, 0    1and k  2.
Special Cases
(i). R 2 ( , 0,1)  R
  
(ii). R 2 ( ,  ,1)  R 

   
14
Preliminary Results
Lemma 2 (Lashin, 2005)[4]
Let p be analytic in E with p(0)  1, p( z )  0 in E and let
arg  p( z )  1 zp '( z ) 

 1 
2

tan 1 11  , 1  0, 1  0.


2
(4)
Then
arg p( z ) 
1
2
for z  E.
15
Main Results
Inclusion result
Theorem 1
For  ,   0,     0, k  2, and 0    1, R k ( ,  ,  )  R k ( , 0,  1 ),
where
 = 1 
2

tan 1 1 1 , 1 

 
(5)
and


 



R k ( ,  ,  )   f  A : 
f ( z ) 
 zf ( z )    P k ( ), z  E  .
 
  



16
Proof
Set
k 1
k 1
f ( z )  p ( z )     p1 ( z )     p2 ( z ) ,
 4 2
 4 2
where p ( z ) is analytic in E with p(0)  1. Then
z  E,




 P k   .
)
z
(
zp

)
z
(
p


 


This implies that




 P   ,
)
z
(
zp

)
z
(
p
i

 i
 


Now with  = 1 
2

tan 1 1 1 and 1 

 
i  1, 2.
, we apply Lemma 2
to have pi  P   1  , i  1, 2, z  E. Consequently, p  P k   1  and
hence, f  R k ( , 0,  1 ).
17
Applications of Theorem 1
Theorem 2
Let f  A and let for   0,   0 and z  E ,
 f ( z) 
f ( z ) 

 z 
 -1

2

 P k   + tan 1




.

(6)
Then

 f ( z) 

  P k   .
 z 
18
Proof

 f ( z) 
Let for  >0, 
  p ( z ). We note that p is analytic
 z 
in E with p (0)  1.Then


1
 f ( z) 


p
(
z
)

zp
(
z
)

f
(
z
)





z




where 1   
2

tan 1
 1
 P k  1  ,

.This implies that, for z  E and i  1, 2,





p
(
z
)

zp
(
z
)
 P  1  ,
i
 i




and, using Lemma 2, we have pi  P   , i  1, 2, z  E.
Therefore, by (2), p  P k 1  and this proves our result.
19
Integral preserving property
For a function f  A given by (1), consider the integral operator I  ,c ,
discussed in [4] and defined by
1
 c

c 1
H ( z )=I  ,c [ f ]( z )=  c  t f (t )dt  ( >0, c  1), z  E. (7)
0
 z

The operator I1,c with c=1,2,..., was introduced by Bernardi. In particular,
z
the operator I1,1 was studied by Libera.
20
Theorem 3
Let f  A and H ( z ) be the integral operator defined by (7).
Let for  >0, c  -1
 f ( z) 
f ( z ) 

 z 
 -1
 H ( z) 
H ( z ) 

 z 
 -1

2
 
 P k   + tan 1
.

 c

Then
 P k   .
21
Convolution properties
Theorem 4
Let f i  R k ( ,  ,  ), i  1, 2 and let  (z )=  f1  f 2  ( z ). Then
 z ( z ) 
 ( z )
 P k ( ), for z  E ,
where  = 1 + and  1 = +
2

tan -1 .
22
Proof.
Since f 2  R k ( ,  ,  ), it follows from Theorem 1, f 2  P k   1  ,
where  1 is given by (5). Similarly f1 P   1  . Let
k 1
k 1
f 2( z )  p( z )     p1 ( z )     p2 ( z ) ,
 4 2
 4 2
f1( z )  h( z ), h, p1 , p2  P   1  .
Now
k 1
k 1
  (h * p1 )( z )     (h * p2 )( z ).
 4 2
 4 2
 ( z )  z ( z )  ( f1* f 2)( z )  
23
Proof (Cont…..)
Applying Lemma 2, we have    P k   , where
1   
Let H ( z ) 
( z ( z )) 
. Then
 ( z )
2

tan 1  .
arg H ( z )  arg( z ( z ))   arg  ( z )

This implies that H  P k    .
k  1
k 
k 
(
) (
)
.
2 2
2 2
4
24
Theorem 5
Let f  R k ( ,  ,  ) and  be a convex univalent function.
Then   f  R k ( ,  ,  ) in E.
Proof.
Let


(( * f )( z ))  
( z ( * f )( z )) ) 
 +
 +

 ( z)  

*
f ( z ) 
( zf ( z ))  
z
 +
  +

 ( z)

* F ( z ),
F  P k ( ).

z
We can write
 ( z)
 k 1   ( z)
  k 1   ( z)

* F ( z)     
* p1 ( z )      
* p2 ( z )  .
z
 4 2  z
  4 2  z

 ( z) 1
Since  is convex, Re

in E , and pi  P( ), i  1, 2.
z
2
25
Proof (Cont…..)
Therefore, by Lemma 1,
 ( z)
* pi ( z ) lies in the
z
connvex hull of pi ( E ). Since pi is analytic in E and
 
 ( z)

pi ( E )     wi : arg wi ( z ) 
.
It
follows
* pi ( z )

2 
z

 ( z)

lies in the . It implies that 
* F ( z )   P k ( ) and
 z

conseqently   f  R k ( ,  ,  ).
26
Applications of Theorem 5
Let f  R k ( ,  ,  ) and fi , i  1, 2,3, 4, also belongs to R k ( ,  ,  ), where
z
f1 ( z )  
0
f (t )
dt , (Alexander integral operator),
t
 1 ( z)   log(1  z)
2
2

(
z
)


[ z  log(1  z )]
2
f 2 ( z )   f (t )dt , (Libera integral operator),
z
z0
z
z
f3 ( z )  
0
f (t )  f (tx)
dt , ( x  1, x  1),
t  tx
 3 ( z) 
1
1  xz
log
1 x
1 x
1 c
f 4 ( z )  c  t c 1 f (t )dt , (Bernardi integral operator).
z 0
z
 4 z) 
1 c n
z

n 11 n  c

27
Radius problem (Inverse inclusion)
Theorem 6


 


R k ( ,  ,  )   f ( z )  A : 
f ( z ) 
 zf ( z)    P k ( ), z  E .
 
  



Let f  R k ( , 0,  ). Then f  R k ( ,  ,  ) for z  r1 , where
r1 
1
21  4  21  1
2
1
,
1 

 
.
(8)
For k  2,   1, we have
Corollary 1
Let f  R. Then f  R ( 1 ) for z  r1 , where
r1 and 1 are given by (8).
Miller and Mocanu [5] proved this result with a different technique.
28
Conclusion
A number of analogous results, in finding connections of the class R k ( ,  ,  )
to the classes C and S * , can be developed by using the techniques given in the book,
The arrow heads show the
inclusion relations.
Differential Subordination by Miller and Mocanu [5].
A
Rk ( ,0,  1 )
R k ( ,  ,  )
2 1

r= 1  tan2 1 1 1 ,, 11 
 
21  41  21  1
1
S
R
S*
1  0
R (1 )
k 2
R k ( ,  ,  )
C
29
References
[1] P.N. Chichra, New subclasses of the class of close-to-convex functions,
Proc. Amer. Math. Soc., 62(1977) 37-43.
[2] A.W. Goodman, Univalent functions, Vol. I, II, Mariner Publishing Company,
Tempa Florida, U.S.A 1983.
[3] J. Krzyz, A counter example concerning univalent functions, Folia Soc. Scient..
Lubliniensis 2(1962) 57-58.
[4] A.Y. Lashin, Applications of Nunokawa's theorem, J. Ineq. Pure Appl. Math., 5(2004),
1-5, Article 111.
[5] S. S. Miller and P. T. Mocanu, Differential subordination theory and applications,
Marcel Dekker Inc., New York, Basel, 2000.
[6] K.I. Noor , On a generalization of alpha convexity, J. Ineq. Pure Appl. Math., 8(2007),
1-4, Article 16.
[7] K.I. Noor and W. Ul-Haq, Some properties of a subclass of analytic functions, Nonlinear
Func. Anal. Appl, 13(2008)265-270.
[8] B. Pinchuk, Functions of bounded boundary rotations, Isr. J. Math., 10(1971),6-16.
[9] S. Singh and R. Singh, Convolution properties of a class of starlike functions, Proc.
Amer. Math. Soc., 106(1989), 145-152.
30
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