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



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



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



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 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



2 2

1

1 1 ( ) ( )lim : , , 0

1z


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



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



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.

The Bloch Space of Analytic functions

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