1) The document discusses Bloch space, which is the space of analytic functions on the unit disk where the supremum of the derivative is finite. It proves properties of Bloch space including that it is invariant under Mobius transformations. 2) It presents two theorems that characterize when an analytic function belongs to Bloch space and little Bloch space. The theorems relate membership to limits of the difference quotient as two points approach each other. 3) It also discusses three containment results, proving relationships between Bloch space, little Bloch space, and the spaces of functions whose derivative is bounded.

1. International
OPEN ACCESS Journal
Of Modern Engineering Research (IJMER)
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 33|
( )
1
w
w z
z for z D
wz


 

  2 1 1 ( )
z D
Sup z f z

  
  2 1 1 ( )
z D
f Sup z f z


 
The Bloch Space of Analytic functions
S. Nagendra1, Prof. E. Keshava Reddy2
1Department of Mathematics, Government Degree College, Porumamilla
2Department of Mathematics, JNTUA
I. INTRODUCTION
We let D
For wD, the Mobius transformation w is defined by
Then
So, the function w maps D on to itself and D on to itself. It is easy to verify that w is its own inverse. Noting
that
 
 
2
1
2
1
( )
1
w
w
z
wz




, the above identity states:
Bloch space B is the space of all analytic functions f on D for which
and B becomes a Banach space with respect to the semi norm
Abstract: We shall state and prove a characterization for the Bloch space and obtain analogous
characterization for the little Bloch space of analytic functions on the unit disk in the complex plane. We
shall also state and prove three containment results related to Bloch space and Little Bloch space.
Keywords: Bloch Space, Analytic Functions, Mobius Transformation
zC / z 1
  
2
2 2
2
2
1 ( ) 1 ( ). ( )
1
1 1
1 1
1 ( ) (1)
1
w w w
w
z z z
w z w z
z wz
w z
z
wz
  


  
     
    
     
 
  

 2 1 2 1 ( ) 1 ( ) (2) w w  z  z    z 

2. The Bloch Space of Analytic functions
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 34|
    2 2
1 1 ( ) ( )
: , ,
1
z w f z f w
f Sup z w D z w
wz z w
      
     
   
 
    2 2
1 1 ( ) ( )
: , ,
1
z w f z f w
Sup z w D z w
wz z w
      
   
   
 
    2 2
1 1 ( ) ( )
, , ,
1
z w f z f w
z w D z w
wz z w
  
     
 
   
 
   
2
2 1
2
2 1
1
1
1
z D
z f z
Sup
z
Sup z f z
f

 
  
 
  
 
   
  
   
   
2 1
2 1
1
1 , (4)
z D
Sup z f z
z f z f z D


   
     
Using (2), we have
     
   
   
2 1
2 1 1
2 1
( )
1
1 ( ) ( )
1 ( ) ( )
(3)
w
w w
z D
w w
z D
w w
z D
w
fo Sup z fo z
Sup z f z z
Sup z f z
f
fo f

 
 
 






 
 
 
 

  
Thus Bloch space is a Mobius invariant space.
In the next section, we shall state and prove a criterion for containment in the Bloch space and little
Bloch space.
II. CHARACTERIZATION FOR BLOCH AND LITTLE BLOCH SPACE
A. THEOREM 1
For an analytic function f on D
Proof : Suppose for an analytic function f on D
Taking limit as wz, we get
For the next part, suppose f 

3. The Bloch Space of Analytic functions
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 35|
1
1
0
f (u) f (0)   f (tu)udt
1
1
0
1
1
0
1
2 2
0
1
0
( ) (0) ( )
( )
( (4))
1
1
f u f f tu udt
f tu u dt
f
u dt
t u
f u
dt
t u


  










Then for each uD, we have
  1
1
0
0
log 1
( ) (0)
1
u t u
f u f f dt f u
t u u  

   
  
  1 1
log 1 log
1
f u f
u

 
  

2
1
log
1
u
f
u 



  2
2
2
2
2
1
1 log 1, 0
1
1 1
1
1 1
1
2
( ) (0) ,
1
u
f x x x
u
u u
f
u
u u
f
u
u
f u f f u D
u




  
       
    
    
  
    
    
  
    
    

Now for z, wD replace f in the above inequality by fow and let u =w (z). Using
w(w(z)) = z and identities (1) and (3) we have
      2
2 ( )
0 .
1 ( )
w
w w w B
w
z
fo u fo fo
z

  

 


4. The Bloch Space of Analytic functions
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 36|
  2 1
1
lim 1 ( ) 0
z
z f z

 
 
 
2
2
1
2
1
1
lim 0
z 1
z
Sup f z
z


 
  
  
  
 
We briefly discuss the little Bloch space B0. The set of all analytic functions f on D for which
For an analytic function f on D and 0 < t < 1 the dilate ft is the function defined by ft(z) = f(tz). It is known that
for an analytic function f on D:
In analogy to theorem (1), we have the following result.
B. THEOREM 2
For an analytic function f on D
    2 2
1
1 1 ( ) ( )
lim : , , 0
z 1
z w f z f w
Sup z w D z w
wz z w 

      
   
   
 
Proof:
Taking limit as wz in the condition of the statement, we get
Suppose f 0, then f - ft
Applying inequality (5) for ft, we have
   2 2
2
2
1
( ) ( )
1 1
1
w z
f
wz
f z f w
w z
wz



  
 

   2 2
1 1 ( ) ( )
2 (5)
1
z w f z f w
f
wz z w 
  
  
 
   2 2
1 1 ( ) ( )
: ,
1 2
,
z w f z f w
z
Sup wz z w f
w D z w

        
      
 
   
0 0 1 t f iff f f as t

   
    2 1
1
0
lim 1 0
z
z f z
f


  
  

5. The Bloch Space of Analytic functions
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 37|
    2 2
1
1 1 ( ) ( )
lim : , , 0
z 1
z w f z f w
Sup z w D z w
wz z w  
      
   
   
 
     
  
2 2 2 2 2
2 2 2 2
1 1 ( ) ( ) 1 1 2 1
.
1 1 1 1
t
t t
z w f z f w z w f t z w t wz
wz z w z w wz t z t w

      

     
Applying inequality (5) for f - ft, we have
        
2 2
1 1
2 (7)
1
t t
t
z w f f z f f w
f f
wz z w 
    
  
 
Inequality (6), (7) and triangle inequality imply that
      
  
         
2 2
2 2
1 1
1
1 1
1
t t t t
z w f z f w
wz z w
z w
f f z f f w f z f w
wz z w
  
 
 
     
 
Now first letting z 1 and then t 1, we get
In the next section, we shall prove three results related to containment of Bloch and Little Bloch space
III. CONTAINMENT RESULTS OF BLOCH AND LITTLE BLOCH SPACE
Let  be a bounded analytic function on D, then there exists a constant
M  0suchthat   z  M,zD
From Cauchy’s integral formula, we have
 
 
 
'
2
1
2 C
w dw
z
i w z






 
  2
2
2
2
1 (6)
1
t
f z
t

  

  
     
  
   
2 2
2 2
1 1
1
1 1
1
t t
t t
z w
f f z f f w
wz z w
z w
f z f w
wz z w
 
   
 
 
 
 
 
  2
2
2
2
2 1
1
t
t
f f f z
t
 
   


6. The Bloch Space of Analytic functions
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 38|
where C is any closed disc of radius r in neighbourhood of 1 and containing z, then   ' 4M
z
r
  in the
concentric disc of radius
2
r
. This implies that  '  z is bounded in any neighbourhood of 1 contained in D
whenever so is   z .
A. THEOREM 3
If f B then f  kB where kC is a constant
Proof: It is very easy to see that
      ' ' f z  f  k z
Therefore           2 ' 2 ' sup 1 sup 1
z D z D
z f z z f k z
 
   
Hence f  kB whenever f B
B. THEOREM 4
If 0 f B
is bounded and  is any bounded and analytic function on D then 0  f B
.
Proof:     2 '
0
1
lim 1 0
z
f B z f z
 
   
Note that
           
           
                 
' ' '
' ' '
2 ' 2 ' 2 ' 1 1 1
f z z f z f z z
f z z f z f z z
z f z z z f z z f z z
  
  
  
 
  
     
Taking limit as
z 1 
, the first term on RHS tends to 0 because of the hypothesis and  is
bounded and the second term since f and
'
bounded in the neighbourhood of 1 as  is bounded on D
tends to 0.
Hence       2 '
1
lim 1 0
z
z  f z
 
 
Therefore 0  f B
.
C. THEOREM 5
If f , g are bounded functions of 0 B , then 0 fgB .
Proof: From the definition of 0 B ,
   
   
2 '
1
2 '
1
lim 1 0
lim 1 0
z
z
z f z
z g z




 
 
Note that                   2 ' 2 ' 2 ' 0  1 z fg z  1 z g z f z  1 z f z g z
Taking limit as
z 1 
, we get
      2 '
1
0
lim 1 0
.
z
z fg z
fg B
 
 
 

7. The Bloch Space of Analytic functions
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 39|
IV. CONCLUSION
I invite interested readers to pursue geometric interpretation of characterization theorems that we proved in this paper and also similar containment results related to the Bloch space.
REFERENCES
[1] J.M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, Proc. of the American Mathematical Society 85,1974, 12-37.
[2] Jose. L. Fernandez, J, On coeffients of Bloch functions, London math. Soc. (2). 29 (1984), 94-102.
[3] R. Aulaskari, N. Danikas, and R. Zhao, The Algebra Property of The Integrals of Some Banach spaces of Analytic functions by N. Danikar, Aristotle univ. of Thessaloniki.
[4] Theory of function spaces by Kehe Jhu.
[5] Bloch functions : The Basic theory by J.M.Anderson edited by S.C. Power, series C: Mathematical and Physical Sciences Vol. 153.
[6] Multipliers of Bloch functions by Jonathan Arazy, Report 54, 1982.
[7] The Bloch space and Besov spaces of Analytic functions by Karel Stroethoff, Bull. Austral. Math. Soc. Vol. 54, 1996, 211-219.