International Journal of Modern Mathematical Sciences, 2012, 4(3): 96-105
International Journal of Modern Mathematical Sciences ISSN:2166-286X
Florida, USA
Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx
Article
A Coefficient Inequality for Certain Classes of Analytic
Functions
B. S. Mehrok1, Gagandeep Singh2,*
1
2
# 643 E, B.R.S. Nagar, Ludhiana (Punjab), India
Department of Mathematics, DIPS College (Co-Educational), Dhilwan(Kapurthala), Punjab, India
* Author to whom correspondence should be addressed; Email: kamboj.gagandeep@yahoo.in
Article history: Received 19 September 2012, Received in revised form 12 November 2012, Accepted
14 November 2012, Published 16 November 2012.
Abstract: We introduce some classes of analytic – univalent functions and for any  (real
or complex) determine the sharp upper bounds of the functional a3  a22 for the

functions of the form f z   z   a k z k belonging to such classes of functions regular in
k 2
the unit disc E   z : z  1 .
Keywords: Univalent functions, Fekete-Szegö functional, starlike functions with respect to
symmetric points, convex functions with respect to symmetric point, close-to-convex
functions with respect to symmetric points.
Mathematics Subject Classification: 30C45
1. Introduction
Let U be the class of bounded functions

wz    c k z k
(1.1)
k 1
which are regular in the unit disc E   z : z  1 and satisfying the conditions w(0)  0 and
wz   1.
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci. 2012, 4(3): 96-105
97
Let A be the class of analytic functions in E of the form

f z   z   a k z k .
(1.2)
k 2
Let S be the class of functions f  z   A and univalent in E.
We recall here the following well known subclasses of A:


2 zf  z 
1  Az
S s  A, B    f  A :

,1  B  A  1, z  E , the
f  z   f  z  1  Bz


functions with
respect to
symmetric points
studied by
subclass of
Goel
starlike
and Mehrok[3]. Also
Ss 1,1  Ss , the class introduced by Sakaguchi[4].


 2zf z   1  Az



K s  A, B    f  A : 
,1  B  A  1, z  E ,
  f z   f  z   1  Bz




the
subclass
of
the subclass
of
convex functions with respect to symmetric points.


 2 zf z   1  Az
 
Cs  A, B    f  A : 
, g  S s ,1  B  A  1, z  E ,
 g z   g  z   1  Bz


close-to-convex functions with respect to symmetric points.
In particular Ks 1,1  Ks , Cs 1,1  Cs , the classes introduced by Das and Singh[1].
For
 1  B  A  1,  1  D  C  1, 0    1, 0    1, and z  E , we consider the following
subclasses of A:

1 





2 zf  z   
2 zf  z 

 1  Az  



C  ; A, B;     f  A : 

 ,

 1  Bz  
 f  z   f  z     f  z   f  z  




1 



 2 zf  z    2 zf  z  


 1  Cz 




Cs  ; A, B; C , D;     f  A : 

,
g

S
A
,
B

,
s







g
z

g

z
1

Dz










g
z

g

z





1 




 2 zf  z    2 zf  z  


 1  Cz 

C1 s   ; A, B; C , D;     f  A : 

, h  K s  A, B .

 1  Dz 
 h z   h z    h z   h z  



s
Fekete and Szegö[2] made an early study for the estimates of a3  a 22 when f  z  is analytic
and univalent in E. The well-known result due to them states that if f  z  is analytic and univalent in
E, then
4  3,   1

  2 

2
a3  a 2  1  2 exp
,0    1 .
1  

3  4 ,   0
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci. 2012, 4(3): 96-105
In
this
paper, we obtain the
98
sharp bounds of
a3  a22 for
the
classes Cs  ; A, B; ,
Cs  ; A, B; C, D;   and C1s   ; A, B; C, D; .
2. Main Result
Theorem 2.1. If f  Cs  ; A, B; , then
(i) For  complex,
 A  B
 21  2  ; 1     1

2
a3  a2   2
2
  A  B     ;     
1
1
 41   2 1
(2.1)
 2  A  B 2
1   ;   1   1

2
 41   
   A  B 
a3  a22  
; 1   1    1   1


2
1

2


 2  A  B 2

  1 ;   1   1
 41   2
(2.2)
(ii) For  real,
where
2

1      1 A  B   2 B    1    A  B 
1 
 1  2  A  B 
1 
and
21   
.
 1  2  A  B 
(2.3)
2
(2.4)
Proof. As f  Cs  ; A, B; , therefore
 2 zf z  


 f z   f  z  
1


 2zf z 


   1  Az  .
  f z   f  z  
 1  Bz 


By definition of subordination, we have
 2 zf z  


 f z   f  z  
1


 2zf z 


   1  Awz   , where w z   U .
 1  Bwz  
  f z   f  z  




(2.5)
An easy calculation gives
1  21   a2 z  21  2 a3   1   ]a22 z 2  ...
   1

 A  B 2 c12  z 2  ...
= 1    A  B c1 z    A  B c2  Bc12  
2


Copyright © 2012 by Modern Scientific Press Company, Florida, USA
(2.6)
Int. J. Modern Math. Sci. 2012, 4(3): 96-105
99
On equating coefficients in (2.6), we get
a2 
 A  B
c1
21   
(2.7)
and
  A  B c2  Bc  
2
1
   1
 2  A  B 2 c12
 A  B  c  2 1   
2
41   
.
21  2 
2
a3 
2
2
1
(2.8)
From (2.7) and (2.8) ,we have
a3  a22 
 A  B
c2
21  2 

 2  A  B 2  1   2  2 B    1 A  B     A  B 1   

  c12 .
2 
 1  2  A  B 
41    

a3  a22 
So
By using
2
a3  a 
  A  B   2  A  B 2
 1     1  c1 2 .

2
21  2  41   
(2.10)
1     1 , then from (2.10), we have
a3  a22 
If
(2.9)
c2  1  c1 , (2.9) becomes
2
2
If
 A  B
 2  A  B 2
c2 
1   c12 .
2
21  2 
41   
 A  B
.
21  2 
1     1 , then again from (2.10), we get
 2  A  B 2
1   .
a3  a 
2
41   
2
2
Now we discuss the case when  is real.
Case I.
For   1 , from (2.10)
  A  B   2  A  B 2
1   1     c1 2 .

a3  a 
2
21  2  41   
2
2
If   1   1 , then (2.11) becomes
 2  A  B 2
1    .
a3  a 
2
41   
2
2
If
1   1    1 , from (2.11)
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
(2.11)
Int. J. Modern Math. Sci. 2012, 4(3): 96-105
100
a3  a22 
Case II.
 A  B
.
21  2 
For   1 , from (2.10)
a3  a22 
  A  B   2  A  B 2
  1   1  c1 2 .

2
21  2 
41   
If   1   1 , then from (2.12)
a3  a22 
 A  B
.
21  2 
If   1   1 , from (2.12)
 2  A  B 2
  1  .
a3  a 
2
41   
2
2
On putting   0 ,   1 in the above theorem , we have the following results:
Corollary 2.1.
(i)
If
f  Ss  A, B , then
For  complex ,
A  B
 2 ; 1     1
2
a3  a 2  
2
A  B    ;     
1
1
1
 4
(ii)
For  real ,
  A  B 2
1   ;   1   1

 4
A  B
a3  a 22  
; 1   1    1   1
 2
  A  B 2
  1 ;   1   1

 4
1 
where
 2B
A  B
and
1 
2
.
 A  B
On putting   1 ,   1 in the Theorem 2.1 , we obtain the following result:
Corollary 2.2.
(i)
If
f  Ks  A, B , then
for  complex ,
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
(2.12)
Int. J. Modern Math. Sci. 2012, 4(3): 96-105
101
A  B
 6 ; 1     1
2
a3  a 2  
2
A  B    ;     
1
1
1
 16
(ii)
for  real ,
  A  B 2
1   ;   1   1

 16
A  B
a3  a 22  
; 1   1    1   1
 6
  A  B 2
  1 ;   1   1

 16
where
1 
 8B
3 A  B 
and
1 
8
.
3 A  B 
Theorem 2.2. If f  Cs  ; A, B; C, D;  , then
(i) for  complex ,
 C  D   A  B 
 31  2   6 ; 2     2

2
a3  a2   2
2
 C  D       A  B  ;     
2
2
2
2

6
 41   
(2.13)
(ii) for  real ,
 2 C  D 2
2      A  B  ;   2   2

2
6
 41   
 C  D   A  B 
a3  a22  

; 2   2    2   2


3
1

2

6

 2 C  D 2

  2    A  B  ;   2   2
2
6
 41   
 1    2  1C  D   4 D   2 1   C  D 
2 
3 1  2 C  D 
(2.14)
2
where
41   
2 
.
3 1  2 C  D 
(2.15)
2
and
(2.16)
Proof. As f  Cs  ; A, B; C, D;  , so
 2 zf z  


 g z   g  z  
1


 2zf z  

   1  Cz  ,
 g z   g  z  
 1  Dz 


Copyright © 2012 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci. 2012, 4(3): 96-105
102

g z   z   bk z k  S s  A, B .
where
(2.17)
k 2
By definition of subordination, we get
 2 zf z  


 g z   g  z  
1


 2zf z  

   1  Cwz   , where w z   U .
 1  Dwz  
 g z   g  z  




(2.18)
An easy calculation gives
1  21   a2 z  1  2 3a3  b3   2 1   ]a22 z 2  ...
   1

= 1   C  D c1 z   C  D c2  Dc12  
C  D 2 c12  z 2  ...
2


(2.19)
On equating coefficients in (2.19), we get
a2 
 C  D 
c1
21   
(2.20)
and
 C  D c2  Dc12  
a3 
   1
2
2 2
2
C  D 2 c12   1     C  D 2 c1  1  2 b3
21   
.
31  2 
(2.21)
From (2.20) and (2.21) ,we have
a3  a22 
 C  D 
c2
31  2 

 2 C  D 2  1   2  4 D  2  1C  D   2  A  B 1   
1

  c12  b3 .

2
3 1  2 C  D 
3
41    

From (2.17), it is easily verified that
2
A  B .
 C  D 
 2 C  D 2
c2 
2   c12 
a3  a 
2
31  2 
6
41   
By using
(2.22)
2
c2  1  c1 , (2.22) becomes
a3  a22 
 C  D   2 C  D 2
 2     2  c1 2   A  B  .

2
6
31  2 
41   
2     2 , then from (2.23), we have
a3  a22 
If
A  B .
2
2
So
If
b3 
 C  D   A  B 

.
6
31  2 
2     2 , then again from (2.23), we get
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
(2.23)
Int. J. Modern Math. Sci. 2012, 4(3): 96-105
a3  a22 
103
A  B .
 2 C  D 2
2   
2
6
41   
Now we discuss the case when  is real.
Case I.
For    2 , from (2.23)
 C  D   2 C  D 2
2   2     c1 2   A  B  .

a3  a 
2
6
31  2 
41   
2
2
(2.24)
If    2   2 , then (2.24) becomes
 2 C  D 2
2      A  B  .
a3  a 
2
6
41   
2
2
If
 2   2     2 , from (2.24)
a3  a22 
 C  D   A  B 

.
6
31  2 
For    2 , from (2.23)
Case II.
 C  D   2 C  D 2
  2   2  c1 2   A  B  .

a3  a 
2
6
31  2 
41   
2
2
(2.25)
If    2   2 , then from (2.25)
a3  a22 
 C  D   A  B 

.
6
31  2 
If    2   2 , from (2.25)
 2 C  D 2
  2    A  B  .
a3  a 
2
6
41   
2
2
On putting   0 ,   1 , A  1, B  1 in the above theorem , we have the following results:
Cor 2.3.
If
(ii)
f  Cs C, D, then
For  complex ,
 C  D   1
; 2     2

3
2
a3  a 2  
2
 C  D      1 ;     
2
2
2

4
3
(ii)
For  real,
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci. 2012, 4(3): 96-105
104
 C  D 2
2     1 ;   2   2

4
3



 C  D 1
a3  a 22  
; 2   2    2   2
3

 C  D 2
  2   1 ;   2   2

4
3

2 
where
 4D
3C  D 
2 
and
4
.
3C  D 
On the same lines, we have the following theorem:
Theorem 2.3. If f  C1s   ; A, B; C, D;  , then
(i) For  complex,
 C  D   A  B 
 31  2   18 ; 3     3

2
a3  a2   2
2
 C  D       A  B  ;     
3
3
3
 41   2
18
(2.26)
(ii) For  real,
 2 C  D 2
3      A  B  ;   3   3

2
18
 41   
 C  D   A  B 
a3  a22  

; 3   3    3   3
18
 31  2 
 2 C  D 2

   3    A  B  ;    3   3
2
18
 41   
where
3 
 1    2  1C  D   4 D   2 1   C  D 
3 1  2 C  D 
(2.27)
2
41   
3 
.
3 1  2 C  D 
(2.28)
2
and
(2.29)
References
[1] R. N. Das and P. Singh, On subclasses of schlicht mappings, Indian J. Pure Appl. Math.,
8(1977): 864-872.
[2] M. Fekete and G. Szegö, Eine Bemer Kung uber ungerade schlichte Functionen , J. London
Math. Soc., 8(1933): 85-89.
Copyright © 2012 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci. 2012, 4(3): 96-105
105
[3] R. M. Goel and Beant Singh Mehrok , A subclass of starlike functions with respect to
symmetric points , Tamkang J. Math., 13(1)(1982): 11-24.
[4] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11(1959): 72-80.
Copyright © 2012 by Modern Scientific Press Company, Florida, USA