International Journal of Modern Mathematical Sciences, 2013, 5(3): 126-132
International Journal of Modern Mathematical Sciences ISSN: 2166-286X
Florida, USA
Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx
Article
Coefficient Estimate for New Subclasses of Analytic Functions
Gagandeep Singh *
Department of Mathematics, DIPS College (Co-Ed.), Dhilwan(Kapurthala), Punjab, India
* Author to whom correspondence should be addressed; E-Mail: kamboj.gagandeep@yahoo.in
Article history: Received 24 December 2012, Received in revised form xx 2013, Accepted xx 2013,
Published xx 2013.
Abstract: For reals A, B, C, D such that 1  D  B  A  C  1 , some new subclasses of analytic

functions f z   z   a k z k in the open unit disc E  {z : z  1} are introduced. In this paper, we shall
k 2
determine the coefficient estimate for functions belonging to these classes.
Keywords: Analytic functions, Close-to-star functions, Subordination, Coefficient estimate.
Mathematics Subject Classification: 30C45
1. Introduction
Let U be the class of bounded functions

w  z    ck z k
k 1
(1)
which are analytic in the unit disc E   z : z  1 and satisfying the conditions
w(0)  0 and w  z   1 , z  E .
Let A denote the class of functions

f  z   z   ak z k
k 2
which are analytic in E .
Let S  and K be the classes of starlike and convex functions respectively.
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A function f  z   A is said to be in the class S  A, B  if it satisfying the following condition
f z  1  Az
,  1  B  A  1, g z   S  , z  E .

g z  1  Bz
(3)
In the sequel, we assume that  1  B  A  1,1  D  C  1, z  E.
T  A, B  is the class of functions f  z   A satisfying the condition
f z  1  Az
, h z   K .

hz  1  Bz
(4)
The classes S  A, B  and T  A, B  are the subclasses of close-to-star functions introduced and
studied by Mehrok et al. in [2] and [3] respectively. Also S 1,1  S , the class of close-to-star
functions introduced by Reade [4].
Now we define some subclasses of functions with S  A, B  and T  A, B .
Definition1.1. Let LS  A, B; C, D be the class consisting of functions f  z   A and satisfying the
condition
zf z 
G  z 

1  Cz
,
1  Dz
(5)
where

G z   z   bk z k  S  A, B .
(6)
k 2
Definition1.2. Let M S  A, B; C, D be the class consisting of functions f  z   A and satisfying the
condition

2zf z 

H z   H  z 

1  Cz
,
1  Dz
(7)
where

H z   z   d k z k  S  A, B  .
(8)
k 2
Definition1.3. LT  A, B; C , D  is the class of functions f  z   A with the condition
zf z 
G  z 

1  Cz
,
1  Dz
(9)
where

Gz   z   bk z k  T  A, B  .
k 2
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Definition1.4. M T  A, B; C , D  is the class of functions f  z   A with the condition

2zf z 

H z   H  z 

1  Cz
,
1  Dz
(11)
where

H z   z   d k z k  T  A, B  .
(12)
k 2
In particular
LS 1,1;1,1  LS , M S 1,1;1,1  M S ,
LT 1,1;1,1  LT , M T 1,1;1,1  M T .
We obtain the coefficient estimate for the above defined classes.
2. Some Preliminary Lemmas
We shall require the following lemmas.
Lemma 2.1.[1] If function Pz  

1  Awz 
 1   pk z k , wz   U ,
1  Bwz 
k 1
then
pn   A  B , n  1 .
(13)
The bounds are sharp , being attained for the functions
P z  
1  Az n
,   1.
1  Bz n
Lemma 2.2.[2] If f  z   S  A, B  , then
 (n  1) A  B 
a n  n 1 
, n  2.
2

(14)
Lemma 2.3.[3] If f  z   T  A, B  , then
an  1  n  1 A  B , n  2.
3. Main Results
Theorem 3.1. Let f z   LS  A, B; C, D then,
an 
C  D   2n  1  n  23n  1 A  B    n.
1  n  1 
n A  B  





n 2 
3 
4


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Proof.
P z  
Let
129

1  Cwz 
 1   p k z k , wz   U , z  E .
1  Dwz 
k 1
Since f z   LS  A, B; C, D , by using (5), we get
zf z   G z Pz , Gz   S  A, B  .
(16)
(17)
Using (6) and (16) in (17), it yields that
1  4a z  9a z
2
3
2
...  n 2 an z n1  ...  1  2b2 z  3b3 z 2 ...  nbn z n1  ...
. 1  p1 z  p2 z 2 ...  pn z n  ... .
(18)
On equating the coefficients of z n 1 in (18), we have
n 2 an  pn1  2b2 pn2  ...  n  1bn1 p1  nbn .
So
n 2 an  pn1  2 b2 pn2  ...  n  1 bn1 p1  n bn .
Using Lemma 2.1 and Lemma 2.2 in (19), we obtain
 n 1

n 2 a n  C  D 1   j b j   n bn
 j 2

 n 1
 A  B  n1 j 3  j 2   n 2 1  n  1 A  B 
 C  D 1   j 2 




2
2


j 2
 j 2

 nn  12n  1  A  B   n 2 n  12 nn  12n  1 




 CD 




6
2
4
6





 n  1 A  B 
 n 2 1 

2



C  D   2n  1  n  23n  1 A  B    n 2
nn  1 
n A  B  



2 
3 
4

Hence
an 
C  D   2n  1  n  23n  1 A  B    n.
1  n  1 
n A  B  



 .

n 2 
3 
4


For A  C  1, B  D  1 , Theorem 3.1 gives the following result for the class LS .
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Corollary 3.2. For f z   LS ,
an 
n2 1
, n  1.
2
f z   M S  A, B; C, D , then
Theorem3.3. If
a 2 n 1 
a2n 
C  D   4n 3  n  n 2  n 6n 2  2n  1 A  B   n A  B   1

6
2n  12  3

C  D   4n 3  n  n 2  n 6n 2  2n  1 A  B .


4n 2
3


6
Proof. Using (7) and (16) in definition 1.2, it gives


2zf z   H z   H  z  Pz , H z   S  A, B  ,
(20)
For (2) and (8), (20) is equal to
1  4a z  9a z ... n a z
2
2
2
3
n 1
n

 ...  4n 2 a2n z 2n1  2n  1 a2n1 z 2n  ...
2
= 1  3d 3 z 2  5d 5 z 4  ...  2n  1d 2n1 z 2n2  ...1  p1 z  p2 z 2  ...  pn z n  ... .
(21)
On equating the coefficients of z 2 n 1 and z 2 n in (21), we get
2n  12 a2n1  p2n  3d 3 p2n2  ...  2n  1d 2n1 p2  2n  1d 2n1 .
4n 2 a2n  p2n1  3d 3 p2n3  ...  2n  1d 2n1 p1 .
(22)
(23)
Applying lemma 2.1 and 2.2 in (22), it gives

n 1


j 1

2n  12 a 2 n 1  C  D 1   2 j  1 d 2 j 1   2n  1 d 2 n 1
n 1
n 1
n 1
n 1

 n 1
 C  D n   j 2  4 j  A  B 4 j 3  4 j 2  

j 1
j 1
j 1
j 1
 j 1
 2n  1 1  n A  B 
 
j
 
2



 4n 3  n n 2  n 6n 2  2n  1  A  B  
 C  D 


6
 3

 2n  1 1  n A  B .
2
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Int. J. Modern Math. Sci. 2013, 5(3): 126-132
131
Again using lemma 2.1 and 2.2 in (23), it yields
 n 1

4n 2 a 2 n  C  D 1   2 j  1 d 2 j 1 
j 1


n

1
n

1
n 1
n 1  

 n1 3


2
2
 C  D n  4 j  4 j  A  B 4 j  4 j  j  

j 1
j 1
j 1
j 1  
 j 1


3
2
2
 4n  n n  n 6n  2n  1  A  B  
 C  D 

 .
6
 3




(25)
Hence from (24) and (25), we have
a 2 n 1 
and
a2n 
C  D   4n 3  n  n 2  n 6n 2  2n  1 A  B   n A  B   1

6
2n  12  3

C  D   4n 3  n  n 2  n 6n 2  2n  1 A  B .
4n 2


3
6


On putting A  C  1, B  D  1 in Theorem 3.3, we obtain the following result for the class M S .
Corollary 3.4. If f z   M S , then
a 2 n 1 

 .  2n  1
2n 2 2n 2  1
2n  1
2
a2n  1 
1
2n 2
Theorem 3.5. Let f  z   LT  A, B; C , D  , then
an 

1  n  1C  D   2n  2 A  B 
1
 n  1 A  B   1.



n
2
3



Proof. Using the technique of Theorem 3.1, this result can be easily proved.
For A  C  1, B  D  1 , Theorem 3.5 gives the following result.
Corollary 3.6. If f  z   LT , then
an 
n  14n  5  2n  1.
3n
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132
Theorem 3.7. Let f  z   M T  A, B; C , D  , then
a 2 n 1 
a2n
C  D  n 2  n 2  n 4n  1 A  B   1  2n A  B 

3
2n  1
2n  12 


C  D   2 n 2  n 4n  1 A  B  

n 
.
4n 2




3
Proof. This result can be easily proved by using the technique of Theorem 3.3.
Putting A  C  1, B  D  1 in Theorem 3.7 , we obtain the following result.
Corollary 3.8. If f  z   M T , then
a 2 n 1


 2 2 n 2  n 4n  1  4n  1

n 

3
2n  12 
 2n  1
a2n 
2
1
2n 2


 2 2 n 2  n 4n  1 
n 
.
3


References
[1]
R. M. Goel and B. S. Mehrok, A subclass of univalent functions, Houston J. Math.,
8(3)(1982) : 343-357.
[2]
B. S. Mehrok, Gagandeep Singh and Deepak Gupta, A subclass of analytic functions, Global J.
Math. Sci.-Th. and Prac., 2(1)(2010): 91-97.
[3]
B. S. Mehrok,
Gagandeep
Singh
and
Deepak
Gupta, On
a
subclass
of
analytic
functions, Antarctica J. Math., 7(4)(2010): 447-453.
[4]
M. O. Reade, On close-to-convex univalent functions, Michigan Math. J., 3(1955): 59-62.
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