Advances in Pure Mathematics, 2012, 2, 291-295
http://dx.doi.org/10.4236/apm.2012.24038 Published Online July 2012 (http://www.SciRP.org/journal/apm)
A Certain Subclass of Analytic Functions
Young Jae Sim, Oh Sang Kwon*
Department of Mathematics, Kyungsung University, Busan, Korea (South)
Received February 20, 2012; revised April 20, 2012; accepted April 28, 2012
ABSTRACT
In the present paper, we introduce a class of analytic functions in the open unit disc by using the analytic function
 

2
3z z  33qz

 , which was investigated by Sokół [1]. We find some properties including the growth
theorem or the coefficient problem of this class and we find some relation with this new class and the class of convex
functions.
Keywords: Univalent Functions; Convex Functions; Subordination; Order of Convexity
1. Introduction
Let H denote the class of analytic functions in the unit
disc

:1zz 

0f
 on the complex plane . Let A
denote the subclass of H consisting of functions normal-
ized by and

00f1

. The set of all func-
tions
f
A

 that are convex univalent in by K. Re-
call that a set E is said to be convex if and only if the
linear segment joining any two points of E lies entirely in
E. Let the function f be analytic univalent in the unit disc
on the complex plane with the normalization.
Then f maps onto a convex domain E if and only if
 

 
0zU







K
Re 1zf z
fz

.
Robertson introduced in [2], the class

of con-
vex functions of order , which is defined by

1 
 

:Re 1,
z
KfA z
fz




 










0, 1  
0
zf  .
If , then a function of this set is univalent
and if



0
it may fail to be univalent. We denote
K
K

. Let S be denote the subset of A which is
composed of univalent functions. We say that f is subor-
dinate to F in , written as
f
F
 
, if and only if,
 
f
zFwz for some Schwarz function


wz ,
and

00w

1wz , z




. The class of convex
functions K can be defined in several ways, for example
we say that f is convex if it satisfies the condition
1
1
zz
1zf
f
zz
 



. (1)
Many subclass of K have been defined by the condi-
tion (1) with a convex univalent function p, given arbi-
trary, instead of the functions 11zz. Janowski
considered the function p, which maps the unit disc onto
a disc in [3,4]. An interesting case when the function p is
convex but is not univalent was considered in [5]. A
function p that is not univalent and is not convex and
maps unit circle onto a concave set was considered in [1].
Now, we shall introduce the class of analytic functions
used in the sequel.
Definition 1.1. The function
f
A belongs to the
class



SQ


, 3,1  

, if it satisfies the condition
 
2
3
33
fz qzzz



 

q
(2)
Let the function

be given by (2). We note that
  


2
31
33 331 3
3
133
qzz zzz
z


 



 

 


 

Sokół investigated in [1] that the image of the unit cir-
cle 1z q under the function


is a curve described
by


2
22 1
:0
2
xax ykx




,
where


   
Reand Im
ii
xqe yqe


,
with


0, 2π  


and 2
91
23
a



 and
*Corresponding author.
C
opyright © 2012 SciRes. APM

Y. J. SIM, O. S. KWON
292


    

2
54
33
k





i
qe
.
Thus the curve is symmetric with respect to real
axis and


satisfies


 

2
91 Re
23
i
qe



3,
23






0, 2π  
(3)
where .
Especially, if 0

, then
 
0
qz , which
maps onto the right of line
11 z
12x
21
31
. And we note
that if

  12 qq , then


.
2. Some Properties of Functions in


SQ

Now we shall find some properties of functions in the
class

SQ

.
Theorem 2.1. If a function f belongs to the class

SQ


,


3,1   , then there exists a function
g
A

such that
 
11
g
zz
hA

and a function such that
 
33 zhz



 
and

f
zgzhz



SQ
.
Proof. Let f be in

. Then there exists an ana-
lytic function with

wz


00w and

1wz

for
such that
z
 
  
3
13wz  
fz wz
. (4)
From (4) we have

    
21
fz wz
f zwz
 
3
wz
wz







.
Define g and h so that


21
g
zwz
g
zwz




and

  
23
wz
wz







hz
hz
 ,
respectively. Then

11
g
zz
,
 
33 zhz


and
222
f
zgzhz
f
zgzhz
 



.







f
zgzhz
Hence 


fSQ
, which proves Theorem
2.1.
Theorem 2.2. If



,

3,1

 and zr,

01r

, then




 

2
2
2
2
1
11 3
1
11 3
fz
rr
rr


 



fSQ
(5)
Proof. Suppose that

. Then



f
zgzhz



For some g and h such that

g
11zz


and

33hz z


,

respectively. And above subordination equations imply
that
 
22
11
11
gz
rr



and
 

 
22
11
13 13
hz
rr




,



respectively. Since
f
zgzhz

, the modulus of


f
z

satisfies the inequality (5).
Next, we shall solve some coefficient problem for a
special function to be in the class

SQ

.

n
n
Theorem 2.3. The function
g
zzcz belongs
to the class


SQ

, whenever

2
2
27 244
43
cn




.


1
1n
nzncz


 
, if we put Proof. Since
g
n
Gzgz

,
then


21
1n Gz ncz


z
.
, Hence for



21Gz nc

.


Re 1Gz nc Since , if

Y. J. SIM, O. S. KWON 293

3
231nc 

 
gz SQ
, (6)
Then n

 and we can easily derive that
the inequality (6) is equivalent to

2
2
24 4
43n
27
c





.
3. The Relations o f the Classes SQ and K
It is well-known that the following implication holds:

 
1
2
fz

 Re 10Re
zf z
fz






 . (7)
More generally, the above implication (7) is can be
generalized as following:

    
zfzzk z
f
zkz



fz kz
 
 .
Evidently, the implication (7) implies the relation

012KSQ. In this chapter, we find some general
relation between the classes


K

and

SQ

.
Let us denote by Q the class of functions f that are
analytic and injective on

Ef


ndlimfz
 
 

, where

Ef  :a
z

and are such that

0
f
Ef


0pa
n


p
.
Lemma 3.1. [6] Let with and let Q

n
a azqz
 Be analytic in with

anda n

Ef
mn
qz .
If q is not subordinate to p, then there exist points
00 and
i
zre

 ,
and there exists a number for which


,qzrp
 
00
qz p


 
m p
and
00
zq z




11
.
Theorem 3.2. Let

 . If a function f belongs
to the class A and


2
Re 13
zf z
fz







forfz


 

fSQ
,
then

.
Proof. Suppose that 0

 and

fSQ



or equiva-
lently,

zqz


0
z
f
.
Then by Lemma 3.1, there exist and



1
,

and m such that 1



0
fz q




and


0
zz
zfz mq






.
Since


23 3
Re 1
12
 



 





,

For

11  ,







 
 

 


 
00
0
Re 1
2
Re 1
33 233
12Re 13 1
23 3
1Re
13
233
2Re1
23 3
1Re
13
3
2Re12
zf z
fz
mq
q
m
m
m
m
m






 


 













 










 


 






 



 





 

 








.
10
In case


, since the inequality (3) induces
the following inequality:


  
2
91 33
Re ,
1323
23


 









(8)








00
0
2
Re 1
23913
121
2
23
91 3
12 1
23 2
231
1,
33
zf z
fz
mm
m






 









 

 





 





 


Y. J. SIM, O. S. KWON
294
which is a contradiction to the hypothesis. In case
01

, using the inequality (8) again,






 
00
0
Re 1
23
12
23
33 3
121
23
313
11
33
zfz
fz
m
m
m

33
1
2
2
21,
3
m






























 


 


 

which is a contradiction to the hypothesis, hence

f
zqz


, and


fSQ

.
If we put 12

 in Theorem 3.2, we can get next
Corollary.
Corollary 3.1. For
f
A, the following implication
holds:

 


2
Re 15
Re
zf z
fz






 1 2
27 .
49
fSQ
fz



  Theorem 3.3. Let and let

1, 1


fSQ

.
Then f is convex for 13z, if 0

, and
236
6
CC
z



,
where

23C3   0, if



fSQ. Proof. Let

. Then
 
2
3
33zz
fz

  

and there exists a Schwarz function


wz with


00w


and

1z s wuch

that
 
2
3
33
fz wzw z

 .
Then






2
4zw z wz
w z 


23
11
33
zf zzw z
fz wz

  

 .
Hence

 


 
23
zf zzw z
fz wz
 


4
13
wz
wz



 (9)
Using the well-known estimate [7]:
 
2
2
1
1
wz
z




wz
,
We have from (9)











2
1234
13
zwz wz
zf z
fz zwz



 

Hence if











2
1234 1
13
zwzwz
zwz



 , (10)
Then f is convex. So it is enough to find the condition
of z0 to satisfy the inequality (10). In case


, then
inequality (10) reduces to



2
21 1
1
zwz
z

, (11)


wz z
And (11) is satisfied for 13z, since .

Hence we can conclude that f is convex for 13z
0
, in
case


. Now we suppose that 0

 and let


wz R

and zr. And let us put








22
2
4234
2333 .
TRrRrrr R
rr



 
Now




2
82340TRrRr rr 


 
implies



2
1
2341 0
8
rr r
RR r




 .


00T

is equivalent to And

2
0
3618
:
3
rr
  


0
rr
0
rr
.
That is, f need not be convex for . And for


0TR

,

is equivalent to


22
2
162333
8
BBr r
RR r


 
 .
where


2
23 4Brrr   

.
Put


242
2
64 16
162333.
PrrBr
rr




Then


048 0P 


and

Y. J. SIM, O. S. KWON
295
REFERENCES
  
3 0.  
22
1 643236432P   
[1] J. Sokół, “A Certain Class of Starlike Functions,” Com-
puters and Mathematics with Applications, Vol. 62, No. 2,
2011, pp. 611-619. doi:10.1016/j.camwa.2011.05.041


1


10Pr 10,r such that Hence there exists a

and for 1 0, . Hence for 1 rr

0Pr0rr


Rr , , attains its maximum at
2
Rr

TR

for
. Now
1
r0Rr


 




2
2
2
0
13 23
323 3
36 ,
6
Tr
rr
rr
CC
r









[2] M. S. Rovertson, “Certain Classes of Starlike Functions,”
Michigan Mathematical Journal, Vol. 76, No. 1, 1954, pp.
755-758.
[3] W. Janowski, “Extremal Problems for a Family of Func-
tions with Positive Real Part and Some Related Families,”
Annales Polonici Mathematici, Vol. 23, 1970, pp. 159-
177.
330
30
r   


[4] W. Janowski, “Some Extremal Problems for Certain Fa-
milies of Analytic Functions,” Annales Polonici Mathe-
matici, Vol. 28, 1973, pp. 297-326.
[5] R. Jurasiska and J. Stankiewics, “Coefficients in Some
Classes Defined by Subordination to Multivalent Majo-
rants,” Annales Polonici Mathematici, Vol. 80, 2003, pp.
163-170.
where 23C3  
, which proves Theorem
3.3.
If we put 12

 in Theorem 3.3, we can get next
Corollary. [6] S. S. Miller and P. T. Mocanu, “Differential Subordina-
tions, Theory and Applications,” Series of Monographs
and Textbooks in Pure and Applied Mathematics, Vol.
225, Marcel Dekker Inc., New York, 2000.

Corollary 3.2. Let

12fSQ. Then f is convex for

173896z 0.453847 .
[7] P. Duren, “Univalent functions, A Series of Comprehen-
sive Studies in Mathematics,” Vol. 259, Springer-Verlag,
New York, 1983.
4. Acknowledgements
The research was supported by Kyungsung University
Research Grants in 2012.