The Bloch Space of Analytic functions
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Transcript of The Bloch Space of Analytic functions
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 zz for z D
wz
2 11 ( )z D
Sup z f z
2 11 ( )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 wD, 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( )
1w
wz
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
/ 1z C z
2
2 2
2
2
1 ( ) 1 ( ). ( )
11 1
1 11 ( ) (1)
1
w w w
w
z z z
w z w z
z wz
w zz
wz
2 211 ( ) 1 ( ) (2)w wz z z
The Bloch Space of Analytic functions
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 34|
2 21 1 ( ) ( )
: , ,1
z w f z f wf Sup z w D z w
z wwz
2 21 1 ( ) ( )
: , ,1
z w f z f wSup z w D z w
z wwz
2 21 1 ( ) ( )
, , ,1
z w f z f wz w D z w
z wwz
22 1
2
2 1
1
1
1z D
z f zSup
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
12
2 1 1
2 1
( )
1
1 ( ) ( )
1 ( ) ( )
(3)
w
w wz D
w wz D
w wz 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 wz, we get
For the next part, suppose f
The Bloch Space of Analytic functions
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 35|
1
1
0
( ) (0) ( )f u f f tu udt
1
1
0
1
1
0
1
220
1
0
( ) (0) ( )
( )
( (4))1
1
f u f f tu udt
f tu u dt
fu dt
t u
f udt
t u
Then for each u D , we have
11
0
0
log 1( ) (0)
1
t uuf u f f dt f u
t u u
1 1
log 1 log1
f u fu
2
1log
1
uf
u
2
2
2
2
2
11 log 1, 0
1
1 1
1
1 1
1
2( ) (0) ,
1
uf x x x
u
u uf
u
u uf
u
uf u f f u D
u
Now for z, w D replace f in the above inequality by fow 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
zfo u fo fo
z
The Bloch Space of Analytic functions
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.11| Nov. 2014 | 36|
2 1
1lim 1 ( ) 0z
z f z
22
1
21
1lim 0
1z
zSup 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
1z
z w f z f wSup z w D z w
z wwz
Proof:
Taking limit as wz 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 zf
wzf z f w
w z
wz
2 21 1 ( ) ( )
2 (5)1
z w f z f wf
z wwz
2 21 1 ( ) ( )
: ,21
,
z w f z f wz
Sup fz wwz
w D z w
0 0 1tf iff f f as t
2 1
1
0
lim 1 0z
z f z
f
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
1z
z w f z f wSup z w D z w
z wwz
2 2 2 2 2
2 22 2
1 1 1 1 2 1( ) ( ).
1 1 1 1
tt t
z w z w f t z w t wzf z f w
z wwz z w wz t z t w
Applying inequality (5) for f - ft, we have
2 2
1 12 (7)
1
t t
t
z w f f z f f wf f
z wwz
Inequality (6), (7) and triangle inequality imply that
2 2
2 2
1 1
1
1 1
1t t t t
z w f z f w
z wwz
z wf f z f f w f z f w
wz z w
Now first letting 1z and then 1t , 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
0 ,M suchthat z M z D
From Cauchy’s integral formula, we have
'
2
1
2C
w dwz
i w z
2
22
21 (6)
1
tf z
t
2 2
2 2
1 1
1
1 1
1
t t
t t
z wf f z f f w
wz z w
z wf z f w
wz z w
2
22
22 1
1t
tf f f z
t
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 ' 4Mz
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 0f B is bounded and
is any bounded and analytic function on D then 0f B
.
Proof: 2 '
01
lim 1 0z
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 1z
, 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
1lim 1 0z
z f z
Therefore 0f B .
C. THEOREM 5
If ,f g are bounded functions of 0B , then 0fg B .
Proof: From the definition of 0B ,
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 1 1z fg z z g z f z z f z g z
Taking limit as
1z , we get
'2
1
0
lim 1 0
.
zz fg z
fg B
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.