CHAPTER 5
COEFFICIENT PROBLEM FOR CERTAIN
CLASSES OF ANALYTIC FUNCTIONS
In this chapter, we established some coefficient problems of the functions belonging to
the various generalized classes of analytic functions. Besides establishing some new
results, we also generalize the results proved by various authors. We also gave a solution
to the well known Rogosinski Conjecture. This chapter has been divided into 6 sections :
Section 5.1: Introduction and Definitions.
Section 5.2: Coefficient Estimate for the class C s  A, B; C , D  .
Section 5.3: The classes C   ; A, B  and C1  ; A, B .
Section 5.4: The classes T   ; A, B  and T1  ; A, B .
Section 5.5: Rogosinski Conjecture.
Section 5.6: Conclusions.
5.1 Introduction and Definitions
For a univalent function in the class A, it is well known that the n-th coefficient is
bounded by n. The bounds for the coefficients give information about the geometric
properties of these functions. For example, the bound for the second coefficient of
normalized univalent functions readily yields the growth and distortion bounds for
116
univalent functions. The coefficient estimates for various classes were discussed by many
authors.
Now we are going to define some important classes as below:
Definition 5.1.1 Let C s  A, B; C , D  be the class consisting of functions f z   A and
satisfying the condition
2 zf  z 
1  Cz

, 1  B  A  1 , 1  D  C  1, z  E ,
g z   g  z  1  Dz
(5.1.1)
where

g  z   z   bk z k  S s*  A, B  .
(5.1.2)
k 2
Obviously Cs 1,1;1,1  Cs , the class of close to convex functions with respect to
symmetric points introduced by Das and Singh [34] and Cs 1,1; C , D   Cs C , D  .
To avoid repetition, throughout this chapter, we assume that  1  B  A  1 , z  E .
Definition 5.1.2 By C   ; A, B   0 , we denote the class of functions f z   A
with f  z   0 and zf z   1    f z   0 in E  0 satisfying

1    f z    zf z   1  Az ,
g z 
g  z  1  Bz
(5.1.3)

g  z   z   bk z k  S  .
where
k 2
In particular,
(i)
C   ;1,1  C    , the class studied by Parvatham and Srinivasan [125].
117
(5.1.4)
C  0; A, B   C   A, B , the subclass of close -to-star
(ii)
functions discussed in
Chapter 4.
(iii)
C  1; A, B   C  A, B ,
the subclass of close -to-convex functions studied
by Mehrok [97].
(iv)
C  0;1,1  C  , the class of close -to-star functions introduced by Reade [130].
(v)
C  1;1,1  C , the class of close-to-convex functions introduced by Kaplan[76].
Definition 5.1.3 Let C1  ; A, B   0 be
the
class of
functions f z   A with
f  z   0 and zf  z   1    f  z   0 in E  0 satisfying

1    f z    zf z   1  Az ,
h z 
h z  1  Bz
(5.1.5)

h z   z   d k z k  K .
where
(5.1.6)
k 2
The following observations are obvious:
(i) C1 0; A, B   C1  A, B , a subclass of close -to-star functions discussed in Chapter 4.
(ii) C1 1; A, B   C1  A, B , a subclass of close-to-convex functions discussed in
Chapter 3.
(iii) C1 1;1,1  C1 , a class introduced by Silverman and Telage [144].
118
Definition 5.1.4 Let T   ; A, B   0 denote the class of functions f z   A which
satisfying the condition
f z
z 2 f z 1 Az
z
, g S  ,


1 Bz
gz
gz
(5.1.7)
There follows the observations:
(i)
T  0; A, B   C  A, B .
(ii) T   ;1,1  T   , the class discussed by Al-Abbadi and Darus [6].
(iii)
T  0;1,1  C.
Definition 5.1.5 Let T1  ; A, B   0 denote the class of functions f z   A which
satisfying the condition
z
f z
z 2 f z 1 Az


, hK .
hz
hz
1 Bz
(5.1.8)
There follows the observations:
(i)
T1 0; A, B   C1  A, B .
(ii) T1 0;1,1  C1.
5.2 Coefficient Estimate for the class C s  A, B; C , D 
In this section, we derived the coefficient estimates for a generalized class of closeto-convex functions with respect to symmetric points. Our result is a generalization of the
119
results due to Das and Singh [34] and Janteng and Halim [73]. This section is based on
the paper “Coefficient estimate for a subclass of close-to-convex functions with
respect to symmetric points” published in “Tamkang
Journal
of
Mathematics”
(China) 2011, Vol. 42, No. 2, pp 217-222.
To prove the main result, we need the following Lemma due to Goel and Mehrok
[48] and Lemma 3.2.1 as discussed in Chapter 3.

Lemma 5.2.1 Let g  z   z   bk z k  S s*  A, B  , then for n  1 ,
k 2
b2 n 
 A  B  n1  A  B  2 j 
n!2 n

j 1
and
b2 n 1 
 A  B  n1  A  B  2 j  .
n!2 n

j 1
Theorem 5.2.1. Let f z   C s  A, B; C , D  , then for n  1 ,
a2n 
a2n1
C  D  n1  A  B  2 j 
n!2 n

(5.2.1)
j 1
n1

A B  

1 
1
2
A
B
j





 C  D 
 .


2n 1
2n   n 1!2n1 j1

(5.2.2)
Proof. By definition of subordination, it follows that f z   C s  A, B; C , D  if and only if
2 zf  z 
1  Cwz 

 P z , w z  U ,
g z   g  z  1  Dwz 
(5.2.3)

where
P  z   1   pk z k .
k 1
120
(5.2.4)
As g  S s*  A, B  , it follws that
2zg   z    g  z   g   z   K  z  for z  E
where
(5.2.5)
K  z   1  d1 z  d 2 z 2  d 3 z 3  ...
On equating the coefficients of like powers of z in (5.2.5), we get
2b2  d1
,
2b3  d 2 ,
4b4  d3  b3 d1 , 4b5  d 4  b3 d 2 ,
(5.2.6)
(5.2.7)
Continuing in this way , we have
2nb2 n  d 2 n 1  b3 d 2 n 3  b5 d 2 n 5  ...  b2 n 1 d1 ,
(5.2.8)
2nb2 n 1  d 2 n  b3 d 2 n  2  b5 d 2 n  4  ...  b2 n 1d 2 .
(5.2.9)
From (5.2.3) and (5.2.4) , we have
z  2a2 z 2  3a3 z 3  ...  2na2 n z 2 n   2n  1 a2 n 1 z 2 n 1  ...
  z  b3 z 3  b5 z 5  ...  b2 n 1 z 2 n 1  b2 n 1 z 2 n 1  ...
. 1  p1 z  p2 z 2  ...  p2 n z 2 n  p2 n 1 z 2 n 1  ... .
(5.2.10)
On equating the coefficients in (5.2.10), we obtain
2a2  p1
3a3  p2  b3 ,
,
4a4  p3  b3 p1 , 5a5  p4  b3 p2  b5 ,
(5.2.11)
(5.2.12)
and so on
2na2 n  p2 n 1  b3 p2 n 3  b5 p2 n 5  ...  b2 n 1 p1 ,
 2n  1 a2 n1  p2 n  b3 p2 n2  b5 p2 n4  ...  b2 n1 p2  b2 n1 .
121
(5.2.13)
(5.2.14)
Using Lemma 3.2.1 and equation (5.2.11) , we get
2 a2  C  D
3 a3   C  D  
,
 A  B .
2
Again applying Lemma 3.2.1 and using equations (5.2.6) and (5.2.7), we obtain from
(5.2.12)
4 a4 
 C  D  A  B  2  ,
5 a5 
2
 A  B  2   A  B   4  C  D 
8
.
It follows that (5.2.1) and (5.2.2) hold for n  1, 2 .
We now prove (5.2.1) and (5.2.2) by induction.
(5.2.13) and (5.2.14) in conjunction with Lemma 3.2.1 yield,
a2n 
C  D  1  n 1 b



2n
k 1
2 k 1



(5.2.15)
and
a2n1 

1 
 n1


1   b2k 1   b2n1  .
C
D




2n 1 
 k 1


(5.2.16)
Again by using Lemma 3.2.1 in (5.2.9) , we have
b2n1
A  B  n1


1 b
2n



k 1
2k 1

.

(5.2.17)
From (5.2.16) and (5.2.17) , we obtain
a 2 n 1 
 A  B  1  n1 b   .
1 




C
D

 2k 1  
2n  1 
2n   k 1

We assume that (5.2.1) and (5.2.2) holds for k  3,4,...,  n 1 .
Using Lemma 5.2.1 in (5.2.15) and (5.2.16), we obtain
122
(5.2.18)
a2n 
C  D  1  n1  A  B  k 1 ( A  B  2 j ) .
2n



k!2 k
k 1



j 1
(5.2.19)
and
a 2 n 1 
 n 1  A  B  k 1

1 
(
A
B
2
j
)


C  D 1  


k
2n  1 
j 1
 k 1 k!2


 A  B  1  n1  A  B  k 1 ( A  B  2 j ) .
2n



k!2 k
k 1


 
j 1
(5.2.20)
In order to prove (5.2.1) , it is sufficient to show that
C  D  1  m1  A  B  k 1 ( A  B  2 j ) = C  D  m 1 ( A  B  2 j ) , (m  3,4,..., n) . (5.2.21)
 


m 
k
2m

k 1
k!2

j 1
m!2
j 1
(5.2.21) is valid for m  3 .
Let us suppose that (5.2.21) is true for all m , 3  m   n  1 . Then from (5.2.19) , we have
C  D  1  n1  A  B  k 1 ( A  B  2 j )
2n



k 1
k!2 k

j 1


 

n  1  C  D   n  2  A  B  k 1
=
(
A
B
2
j
)



1  


n  2n  1  k 1 k!2 k j 1
 

C  D  .  A  B  n2 ( A  B  2 j )
n  1!2 n1 
2n
j 1
=
n  1 . C  D  n2 ( A  B  2 j )

n n  1!2 n 1 j 1

C  D  .  A  B  n2 ( A  B  2 j )
n  1!2 n1 
2n
j 1
=
C  D  n2 ( A  B  2 j )  A  B  2n  1
2n
n  1!2 n1 
j 1
123
=
C  D  n1 ( A  B  2 j ).

n!2 n
j 1
Thus (5.2.21) holds for m  n and hence (5.2.1) follows.
Now from (5.2.20) , we have
a 2 n 1 
 A  B  1  n1  A  B  k 1  A  B  2 j   .
1 
C  D  
 


2n  1 
2n   k 1 k!2 k j 1
 
(5.2.22)
From (5.2.21) , we have
n 1
1 
k 1
 A  B  k 1 ( A  B  2 j ) =

k
k!2
j 1
n 1
1
( A  B  2 j)
n  1!2 n1 
j 1
(5.2.23)
From (5.2.22) and (5.2.23), we have
a2n1 
 A B   1 n1 A B  2 j 
1 
 
 C  D 


2n 1
2n   n 1 !2n1 j1

which proves (5.2.2).
Putting A  C  1 and B  D  1 in the above result , we get the following result:
Corollary 5.2.1 Let f z  be schlicht and starlike with respect to symmetric points in the

unit disc E , having the form f z   z   ak z k , then
k 2
an  1 for any natural number n .
This result was proved by Das and Singh [34].
For C  1 and D  1 , we have the following result for the class C s  A, B  .
124
Corollary 5.2.2 Let f  C s  A, B  , then for n  1 ,
a2n1 
n1

1   A B  
1
A
B
j


2


2 

 .


2n 1
2n   n 1 !2n1 j1

Remark- This result was proved by Janteng and Halim [73], but the result given by them
is not justified.
5.3 The classes C   ; A, B  and C1  ; A, B 
In this section, we obtained the coefficient estimate and integral representation


formulae for the functions in the classes C  ; A, B  and C1  ; A, B . The results due to
Parvatham and Srinivasan [125], Reade [130], Mehrok [97] and Silverman and Telage
[144] follows as special cases from our results. Also some of the results proved in
Chapter 3 and Chapter 4 are particular cases of these results.
5.3.1 Integral Representation Formula for the class C   ; A, B 
Theorem 5.3.1 A function f z  is in C   ; A, B  if and only if there exists a starlike
function g z  and a function Pz  which is analytic and has a positive real part in E such
that
f z  
z
1
2
1
t  Pt g t dt
 1 
z 0
 Pz g z 
for   0.
for   0.
Proof. As f  z   C   ; A, B   0, then
1    f z   zf z   Pz g z ,
125
(5.3.1)
where g z   S  and Pz  is a function with positive real part in E.
1
2
1 
If   0, then multiply (5.3.1) by  z
1
and integrating with respect to z , we get
1
z  f z  
1

1
z
2
 t  Pt g t dt
0
or
f z  
On choosing g z  
z
1
2
1
t  Pt g t dt.
 1 
z 0
z
1  Az
and P z  
in the above result, we obtain:
2
1  Bz
1  z 
1
f z  
z
1
1
z 
1
1
t  1  At 
 1  t  1  Bt  dt
2
for   0.
0
If   0, then f  z   P z g z .
The converse is immediate.
5.3.2 Coefficient Estimate for the class C   ; A, B 
Theorem 5.3.2 If f  z   C   ; A, B  , then
 2  (n  1) A  B  
an  n
, n  2 .
 2n  1    
(5.3.2)
Proof. As f  z   C   ; A, B  , then by definition of subordination


1    f z    zf z   1  Awz   1   pk z k , wz   U .
g z 
g  z  1  Bw z 
k 1
Using (1.2.1) and (5.1.4) in (5.3.3), we get
1   z  a 2 z 2  a3 z 3  ...  a n z n  ...  z 1  2a 2 z  3a3 z 2  ...  na n z n 1  ...
126
(5.3.3)



 z  b2 z 2  b3 z 3  ...  bn z n  ... 1  p1 z  p 2 z 2  ...  p n 1 z n 1  ... .
(5.3.4)
On equating the coefficients of z n in (5.3.4), we have
1    n a n
 p n 1  b2 p n  2  ...  bn 1 p1  bn .
Using the fact bn  n for n  2 and Lemma 3.2.1, we get
1    n  a n
  A  B 1  2  ...  n  1  n.
Hence the Theorem 5.3.2.
The bounds are sharp being attained for the function obtained in Theorem 5.3.1.
For A  1, B  1 ,Theorem 5.3.2 gives the following result due to Parvatham and
Srinivasan [125].
Corollary 5.3.1 If f z  C    , then
an 
n2
, n  2.
n  1   
On putting   0, Theorem 5.3.2 yields
Corollary 5.3.2 If f z   C   A, B  , then
 (n  1) A  B  
a n  n 1 
, n  2.
2

For   1, Theorem 5.3.2 gives the following result proved by Mehrok [97].
Corollary 5.3.3 If f z   C  A, B  , then
an  1 
(n  1) A  B 
, n  2.
2
On putting   0 and A  1, B  1 Theorem 5.3.2 observes the result due to Reade
[130].
127
Corollary 5.3.4 If f z  C  , then
a n  n 2 , n  2.
5.3.3 Integral Representation Formula for the class C1  ; A, B 
Theorem 5.3.3 A function f z  is in C1  ; A, B  if and only if there exists a convex
function hz  and a function Pz  which is analytic and has a positive real part in E such
that
f z  
1
z
2
1
t  Pt ht dt
 1 
z 0
 Pz h z 
for   0,
for   0.
Proof. The proof of this theorem is similar to Theorem 5.3.1.
On choosing hz  
z
1  Az
and P z  
in the above result, we obtain:
1  Bz
1  z 
1
f z  
1
1
z 
1
1
t 1  At 
0 1  t 1  Bt  dt for   0.
z

5.3.4 Coefficient Estimate for the class C1  ; A, B 
Theorem 5.3.4 If f  z   C1  ; A, B  , then
an 
1  (n  1) A  B 
, n  2.
n  1  
Proof. Using the technique of Theorem 5.3.2, we can easily prove this theorem.
Equality holds for the functions obtained in Theorem 5.3.3.
Putting   0, Theorem 5.3.4 follows:
128
Corollary 5.3.5 If f z   C1  A, B  , then
a n  1  (n  1) A  B , n  2.
For   1, Theorem 5.3.4 gives the following result.
Corollary 5.3.6 If f z   C1  A, B  , then
an 
1 (n  1) A  B 

, n  2.
n
n
For   1 and A  1, B  1 Theorem 5.3.4 yields the result of Silverman and Telage
[144].
Corollary 5.3.7 If f z   C1 , then
1
a n  2  , n  2.
n
5.4 The classes T   ; A, B  and T1  ; A, B 
This section is concerned
with
the integral representation
formulae
and
coefficient estimates for the functions in the classes T   ; A, B  and T1  ; A, B . The
results due to Mehrok [97] and Silverman and Telage [144] are particular cases of our
theorems.
5.4.1 Integral Representation Formula for the class T   ; A, B 
Theorem 5.4.1 A function f z  is in T   ; A, B  if and only if there exists a starlike
function g z  and a function Pz  which is analytic and has a positive real part in E such
that
129
t

1  c  c1

u g u .Pu du dt
c1  
t 0
0

z
f z   
where c 
1

(5.4.1)
 1,   0.
 g t  
f z    
 Pt dt.
t 
0
z
If   0 , then
(5.4.2)
Proof. Let f z   T   ; A, B   0, then
zf  z 
z 2 f  z 

 Pz  ,
g z 
g z 
(5.4.3)
where Pz  is a function with positive real part in E.

zf  z 
z  zf  z 
1   
 P  z .

g z 
g z 
(5.4.3) is equivalent to
Dividing by  and putting c 
1

 1, we get

zf  z  z  zf  z 
c
 1  c P  z .

g z 
g z 
(5.4.4)
Multiplying (5.4.4) by z c 1 g  z  , we obtain

cz c1  zf  z   z c  zf  z   1  c z c1 g  z P z 
which implies
zf z .z   1  c z g z Pz .
c
c 1
Therefore on integrating (5.4.5) with respect to z , we obtain (5.4.1).
Conversely, if f z  satisfies (5.4.1), then it is easy to see that f z   T   ; A, B .
130
(5.4.5)
On choosing g z  
z
1  Az
and Pz  
2
1  Bz
1  z 
in (5.4.1) , we obtained the following
function of T   ; A, B  :
z
f  z    1  c t
0
 c 1

 t u c 1  Au 
1 

du  dt ,  c   1 .

2
 

 0 1  u  1  Bu  
5.4.2 Coefficient Estimate for the class T   ; A, B 
Theorem 5.4.2 Let f  z   T   ; A, B  , then
 2  (n  1) A  B  
an  
, n  2 .
 2n  1    
(5.4.6)
Proof. Since f  z   T   ; A, B  , then by definition of subordination

f z
z 2 f z 1  Aw z 
z


 1   pk z k , w z   U .
1  Bw z 
gz
gz
k 1
(5.4.7)
Using (1.2.1) and (5.1.4) in (5.4.7), we get




z 1  2a 2 z  3a 3 z 2  ...  na n z n 1  ...  z 2 2a 2  6a 3 z  ...  nn  1a n z n  2  ...



 z  b2 z 2  b3 z 3  ...  bn z n  ... 1  p1 z  p 2 z 2  ...  p n 1 z n 1  ... .
(5.4.8)
On equating the coefficients of z n in (5.4.8), we have
n1    n a n  p n 1  b2 p n  2  ...  bn 1 p1  bn .
Using the fact bn  n for n  2 and Lemma 3.2.1, we get
n1    n  a n   A  B 1  2  ...  n  1  n.
This proves the Theorem 5.4.2.
The bounds are sharp being attained for the function obtained in Theorem 5.4.1.
For   0, Theorem 5.4.2 leads to the result of Corollary 5.3.3.
131
5.4.3 Integral Representation Formula for the class T1  ; A, B 
Theorem 5.4.3 A function f z  is in T1  ; A, B  if and only if there exists a convex
function hz  and a function Pz  which is analytic and has a positive real part in E such
that
t

1  c  c1

f z    c1   u hu .Pu du dt
t 0
0

z
where c 
1

 1,   0.
 ht  
f z    
 Pt dt.
t 
0
z
If   0 , then
Proof. The proof of this theorem is similar to Theorem 5.4.1.
On choosing hz  
z
1  Az
and P z  
in the above result, we obtain the following
1  z 
1  Bz
z
f  z    1  c t
0
 c 1
 t u c 1  Au 

1 

du  dt ,  c   1 .

 

 0 1  u 1  Bu  
5.4.4 Coefficient Estimate for the class T1  ; A, B 
Theorem 5.4.4 Let f  z   T1  ; A, B  , then
an 
1  (n  1) A  B 
, n2 .
nn  1   
Proof. Using the technique of Theorem 5.4.2, we can easily prove this theorem.
Equality holds for the functions obtained in Theorem 5.4.3.
For   0, Theorem 5.4.4 leads to the result of Corollary 5.3.6.
132
5.5 Rogosinski Conjecture
In this section, we established a solution to the well known conjecture due to
Rogosinski. This section is based on the paper “A solution of Rogosinski Conjecture”
published in “Antarctica Journal of Mathematics” (India) 2011, Vol.8, No. 1, pp 37-40.
In 1916, Bieberbach [21] made the following conjecture:

If f z   z   a k z k is analytic and univalent in the unit disc E  z : z  1, then
k 2
an  n .
Rogosinski [133] introduced the concept of subordination and conjectured:
If


k 2
k 2
f  z   z   a k z k is univalent in E and g  z   z   bk z k be analytic in E such that
g  z   f  z  , then
bn  n .
This result is known as Rogosinski Conjecture.
To prove this result, we need the following result due to Rogosinski [133]:
Lemma 5.5.1 (Rogosinski Inequality)


n 0
n 0
If f z    a n z n  F z    An z n , then
n
 ak
k 0
Theorem 5.5.1 If
2
n
  Ak .
2
k 0


k 2
k 2
f  z   z   a k z k is univalent in E and g  z   z   bk z k be analytic
in E such that g z   f z  , then bn  n .
133
Proof. Using Lemma 5.5.1, we have
n
 bk
2
k 1
n
  ak
2
k 1
which gives
b1  b2  b3  ...  bn
2
2
2
2
 a1  a 2  a3  ...  a n .
2
2
2
2
Using Bieberbach conjecture, we have
b1  b2  b3  ...  bn
2
2
2
 12  2 2  3 2  ...  n 2
2
=
nn  12n  1
.
6
(5.5.1)
Similarly
b1  b2  b3  ...  bn 1 
2
2
2
2
n  1n2n  1 .
6
(5.5.2)
Subtracting (5.5.2) from (5.5.1) , we have
bn
2
 n2 .
Hence
bn  n .
5.6 Conclusions
The results derived in this chapter are very useful. In Section 5.2, the generalized
class of close-to- convex functions with respect to symmetric points has been investigated
and coefficient estimate for this class has been established. By giving particular values to
the parameters A, B, C and D, one can easily obtain the results proved earlier by Das and
134
Singh [34] and Janteng and Halim [73]. Also in Sections 5.3 and 5.4, we investigated the
subclasses of Alpha- close-to-star functions and obtained sharp coefficient estimate and
extremal functions. The results already proved by Parvatham and Srinivasan [125],
Mehrok [97], Silverman and Telage [144] and Reade [130] are particular cases of our
results. In Section 5.5, a solution to the well known conjecture made by Rogosinski
[133], has been established.
135