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Natural metrics and composition operators in generalized hyperbolic function spaces

Journal of Inequalities and Applications20122012:185

https://doi.org/10.1186/1029-242X-2012-185

  • Received: 29 March 2012
  • Accepted: 31 May 2012
  • Published:

Abstract

In this paper, we define some generalized hyperbolic function classes. We also introduce natural metrics in the generalized hyperbolic ( p , α ) -Bloch and in the generalized hyperbolic Q ( p , s ) classes. These classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, boundedness and compactness the composition operators C ϕ acting from the generalized hyperbolic ( p , α ) -Bloch class to the class Q ( p , s ) are characterized by conditions depending on an analytic self-map ϕ : D D .

MSC:47B38, 46E15.

Keywords

  • hyperbolic classes
  • composition operators
  • ( p , α ) -Bloch space
  • Q ( p , s ) classes

1 Introduction

Let D = { z : | z | < 1 } be the open unit disc of the complex plane C , D its boundary. Let H ( D ) denote the space of all analytic functions in D and let B ( D ) be the subset of H ( D ) consisting of those f H ( D ) for which | f ( z ) | < 1 for all z D . Also, d A ( z ) be the normalized area measure on D so that A ( D ) 1 . The usual α-Bloch spaces B α and B α , 0 are defined as the sets of those f H ( D ) for which
and

respectively. Now, we will give the following definition:

Definition 1.1 The ( p , α ) -Bloch spaces B p , α and B p , α , 0 are defined as the sets of those f H ( D ) for which
and

where 0 < p , α < .

Remark 1.1 The definition of ( p , α ) -Bloch spaces is introduced in the present paper for the first time. One should note that, if we put p = 2 in Definition 1.1, we will obtain the spaces B α and B α , 0 .

Remark 1.2 ( p , α ) -Bloch space is very useful in some calculations in this paper and it can be also used to study some other operators like integral operators (see [12]).

If ( X , d ) is a metric space, we denote the open and closed balls with center x and radius r > 0 by B ( x , r ) : = { y X : d ( y , x ) < r } and B ¯ ( x , r ) : = { y X : d ( y , x ) = r } , respectively. The well-known hyperbolic derivative is defined by f ( z ) = | f ( z ) | 1 | f ( z ) | 2 of f B ( D ) and the hyperbolic distance is given by ρ ( f ( z ) , 0 ) : = 1 2 log ( 1 + | f ( z ) | 1 | f ( z ) | ) between f ( z ) and zero.

A function f B ( D ) is said to belong to the hyperbolic α-Bloch class B α if
f B α = sup z D f ( z ) ( 1 | z | 2 ) α < .
The little hyperbolic Bloch-type class B α , 0 consists of all f B α such that
lim | z | 1 f ( z ) ( 1 | z | 2 ) α = 0 .

The Schwarz-Pick lemma implies B α = B ( D ) for all α 1 with f B α 1 , and therefore, the hyperbolic α-Bloch classes are of interest only when 0 < α < 1 .

It is obvious that B α is not a linear space since the sum of two functions in B ( D ) does not necessarily belong to B ( D ) .

Now, let 0 < p < , we define the hyperbolic derivative by f p ( z ) = p 2 | f ( z ) | p 2 1 | f ( z ) | 1 | f ( z ) | p of f B ( D ) . When p = 2 , we obtain the usual hyperbolic derivative as defined above.

A function f B ( D ) is said to belong to the generalized hyperbolic ( p , α ) -Bloch class B p , α if
f B p , α = sup z D f p ( z ) ( 1 | z | 2 ) α < .
The little generalized ( p , α ) -hyperbolic Bloch-type class B p , α , 0 consists of all f B p , α such that
lim | z | 1 f p ( z ) ( 1 | z | 2 ) α = 0 .
Let the Green’s function of D be defined as g ( z , a ) = log 1 | φ a ( z ) | , where φ a ( z ) = a z 1 a ¯ z is the Möbius transformation related to the point a D . For 0 < p , s < , the hyperbolic class Q ( p , s ) consists of those functions f B ( D ) for which
f Q ( p , s ) p = sup a D D ( f p ( z ) ) 2 g s ( z , a ) d A ( z ) < .
Moreover, we say that f Q ( p , s ) belongs to the class Q ( p , s , 0 ) if
lim | a | 1 D ( f p ( z ) ) 2 g s ( z , a ) d A ( z ) = 0 .

When p = 2 , we obtain the usual hyperbolic Q class as studied in [10, 11, 14].

Remark 1.3 The Schwarz-Pick lemma implies that B p , α = B ( D ) for all α 1 with f B p , α 1 and therefore, the generalized hyperbolic ( p , α ) -classes are of interest only when 0 < α < 1 . Also Q ( p , s ) = B ( D ) for all s > 1 , and hence, the generalized hyperbolic Q ( p , s ) -classes will be considered when 0 s 1 .

For any holomorphic self-mapping ϕ of D , the symbol ϕ induces a linear composition operator C ϕ ( f ) = f ϕ from H ( D ) or B ( D ) into itself. The study of a composition operator C ϕ acting on the spaces of analytic functions has engaged many analysts for many years (see, e.g., [18, 11, 13, 16] and others).

Yamashita was probably the first to consider systematically hyperbolic function classes. He introduced and studied hyperbolic Hardy, BMOA and Dirichlet classes in [1820] and others. More recently, Smith studied inner functions in the hyperbolic little Bloch-class [15], and the hyperbolic counterparts of the Q p spaces were studied by Li in [10] and Li et al. in [11]. Further, hyperbolic Q p classes and composition operators were studied by Pérez-González et al. in [14].

In this paper, we will study the generalized hyperbolic ( p , α ) -Bloch classes B p , α and the hyperbolic Q ( p , s ) type classes. We will also give some results to characterize Lipschitz continuous and compact composition operators mapping from the generalized hyperbolic ( p , α ) -Bloch class B p , α to Q ( p , s ) classes by conditions depending on the symbol ϕ only. Thus, the results are generalizations of the recent results of Pérez-González, Rättyä and Taskinen [14].

Recall that a linear operator T : X Y is said to be bounded if there exists a constant C > 0 such that T ( f ) Y C f X for all maps f X . By elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the Lipschitz-continuity. Moreover, T : X Y is said to be compact if it takes bounded sets in X to sets in Y which have compact closure. For Banach spaces X and Y contained in B ( D ) or H ( D ) , T : X Y is compact if and only if for each bounded sequence ( x n ) X , the sequence ( T x n ) Y contains a subsequence converging to a function f Y .

Throughout this paper, C stands for absolute constants which may indicate different constants from one occurrence to the next.

The following lemma follows by standard arguments similar to those outlined in [17]. Hence we omit the proof.

Lemma 1.1 Assume ϕ is a holomorphic mapping from D into itself. Let 0 < p , s < , and 0 < α < . Then C ϕ : B p , α Q ( p , s ) is compact if and only if for any bounded sequence ( f n ) n N B p , α which converges to zero uniformly on compact subsets of D as n , we have lim n C ϕ f n Q ( p , s ) = 0 .

Using the standard arguments similar to those outlined in Lemma 1 of [9], we have the following lemma:

Lemma 1.2 Let 0 < α < , then there exist two functions f , g B p , α such that for some constant C,
( | f p ( z ) | + | g p ( z ) | ) ( 1 | z | 2 ) α C > 0 , for each  z D .

2 Natural metrics in B p , α and Q ( p , s ) classes

In this section we introduce natural metrics on generalized hyperbolic α-Bloch classes B p , α and the classes Q ( p , s ) .

Let 0 < p , s < , and 0 < α < 1 . First, we can find a natural metric in B p , α (see [14]) by defining
d ( f , g ; B p , α ) : = d B p , α ( f , g ) + f g B p , α + | f ( 0 ) g ( 0 ) | p 2 ,
(1)
where
d B p , α ( f , g ) : = sup z D | f ( z ) | f ( z ) | p 2 1 1 | f ( z ) | p g ( z ) | g ( z ) | p 2 1 1 | g ( z ) | p | ( 1 | z | 2 ) α .
For f , g Q ( p , s ) , define their distance by
d ( f , g ; Q ( p , s ) ) : = d Q ( f , g ) + f g Q ( p , s ) + | f ( 0 ) g ( 0 ) | p 2 ,
where
d Q ( f , g ) : = ( p 2 sup z D D | f ( z ) | f ( z ) | p 2 1 1 | f ( z ) | p g ( z ) | g ( z ) | p 2 1 1 | g ( z ) | p | 2 g s ( z , a ) d A ( z ) ) 1 2 .

Now, we give a characterization of the complete metric space d ( , ; B p , α ) .

Proposition 2.1 The class B p , α equipped with the metric d ( , ; B p , α ) is a complete metric space. Moreover, B p , α , 0 is a closed (and therefore complete) subspace of B p , α .

Proof Clearly d ( f , g ; B p , α ) 0 , d ( f , g , B p , α ) = d ( g , f ; B p , α ) . Also,
d ( f , h ; B p , α ) d ( f , g ; B p , α ) + d ( g , h ; B p , α ) .

Moreover, d ( f , f ; B p , α ) = 0 for all f , g , h B p , α .

It follows from the presence of the usual ( p , α ) -Bloch term that d ( f , g ; B p , α ) = 0 implies f = g . Hence, ( B p , α , d ) is a metric space. Let ( f n ) n = 1 be a Cauchy sequence in the metric space ( B p , α , d ) , that is, for any ε > 0 , there is an N = N ( ε ) N such that
d ( f n , f m ; B p , α ) < ε
for all n , m > N . Since ( f n ) B ( D ) , the family ( f n ) is uniformly bounded and hence normal in D . Therefore, there exist f B ( D ) and a subsequence ( f n j ) j = 1 such that f n j converges to f uniformly on compact subsets, and by the Cauchy formula, the same also holds for the derivatives. Let m > N . Then the uniform convergence yields
(2)
for all z D , and it follows that
f B p , α f m B p , α + ε .

Thus, f B p , α as desired. Moreover, (2) and the completeness of the usual ( p , α ) -Bloch imply that ( f n ) n = 1 converges to f with respect to the metric d. The second part of the assertion follows by (2). □

Next, we give a characterization of the complete metric space d ( , ; Q ( p , s ) ) .

Proposition 2.2 The class Q ( p , s ) equipped with the metric d ( , ; Q ( p , s ) ) is a complete metric space. Moreover, Q ( p , s , 0 ) is a closed (and therefore complete) subspace of Q ( p , s ) .

Proof For f , g , h Q ( p , s ) , then clearly

  • d ( f , g ; Q ( p , s ) ) 0 ,

  • d ( f , f ; Q ( p , s ) ) = 0 ,

  • d ( f , g ; Q ( p , s ) ) = 0 implies f = g ,

  • d ( f , g ; Q ( p , s ) ) = d ( g , f ; Q ( p , s ) ) ,

  • d ( f , h ; Q ( p , s ) ) d ( f , g ; Q ( p , s ) ) + d ( g , h ; Q ( p , s ) ) .

Hence, d is metric on Q ( p , s ) .

For the completeness proof, let ( f n ) n = 1 be a Cauchy sequence in the metric space ( Q ( p , s ) , d ) , that is, for any ε > 0 there is an N = N ( ε ) N such that d ( f n , f m ; Q ( p , s ) ) < ε , for all n , m > N . Since f n B ( D ) such that f n converges to f uniformly on compact subsets of D . Let m > N and 0 < r < 1 . Then Fatou’s lemma yields
and by letting r 1 , it follows that
D ( f p ( z ) ) 2 g s ( z , a ) d A ( z ) 2 ε 2 + 2 D | | f m ( z ) | p 2 1 f m ( z ) 1 | f m ( z ) | p | 2 g s ( z , a ) d A ( z ) .
(3)
This yields
f Q ( p , s ) p 2 ε 2 + 2 f m Q ( p , s ) 2 ,

and thus f Q ( p , s ) . We also find that f n f with respect to the metric of Q ( p , s ) . The second part of the assertion follows by (3). □

3 Lipschitz continuous and compactness of C ϕ

Theorem 3.1 Let 0 < p < , 0 s 1 , and 0 < α 1 . Assume that ϕ is a holomorphic mapping from D into itself. Then the following statements are equivalent:
  1. (i)

    C ϕ : B p , α Q ( p , s ) is bounded;

     
  2. (ii)

    C ϕ : B p , α Q ( p , s ) is Lipschitz continuous;

     
  3. (iii)

    sup a D D | ϕ ( z ) | 2 ( 1 | ϕ ( z ) | p ) 2 α g s ( z , a ) d A ( z ) < .

     
Proof First, assume that (i) holds, then there exists a constant C such that
C ϕ f Q ( p , s ) C f B p , α , for all  f B p , α .
For given f B p , α , the function f t ( z ) = f ( t z ) , where 0 < t < 1 , belongs to B p , α with the property f t B p , α f B p , α . Let f, g be the functions from Lemma 1.2 such that
1 ( 1 | z | 2 ) α | f p ( z ) | + | g p ( z ) | ,
for all z D , so that
| ϕ ( z ) | ( 1 | ϕ ( z ) | ) α ( f ϕ ) ( z ) + ( g ϕ ) ( z ) .
Thus,

This estimate together with the Fatou’s lemma implies (iii).

Conversely, assuming that (iii) holds and that f B p , α , we see that

Hence, it follows that (i) holds.

(ii) (iii). Assume first that C ϕ : B p , α Q ( p , s ) is Lipschitz continuous, that is, there exists a positive constant C such that
d ( f ϕ , g ϕ ; Q ( p , s ) ) C d ( f , g ; B p , α ) , for all  f , g B p , α .
Taking g = 0 , this implies
f ϕ Q ( p , s ) C ( f B p , α + f B p , α + | f ( 0 ) | p 2 ) , for all  f B p , α .
(4)
The assertion (iii) for α = 1 follows by choosing f ( z ) = z in (4). If 0 < α < 1 , then
2 p | f ( z ) | p 2 = | 0 z | f ( t ) | p 2 f ( t ) d t + ( f ( 0 ) ) p 2 | f B p , α 0 | z | d x ( 1 x 2 ) α + | f ( 0 ) | p 2 f B α ( 1 α ) + | f ( 0 ) | p 2 ,
this yields
2 p | f ( ϕ ( 0 ) ) g ( ϕ ( 0 ) ) | p 2 f g B p , α ( 1 α ) + | f ( 0 ) g ( 0 ) | p 2 .
Moreover, Lemma 1.2 implies the existence of f , g B p , α such that
| f p ( z ) + g p ( z ) | ( 1 | z | 2 ) α C > 0 , for all  z D .
(5)
Combining (4) and (5), we obtain

for which the assertion (iii) follows.

Assume now that (iii) is satisfied, we have
d ( f ϕ , g ϕ ; Q ( p , s ) ) = d Q ( p , s ) ( f ϕ , g ϕ ) + f ϕ g ϕ Q ( p , s ) + | f ( ϕ ( 0 ) ) g ( ϕ ( 0 ) ) | p 2 d B p , α ( f , g ) ( sup a D D | ϕ ( z ) | 2 ( 1 | ϕ ( z ) | p ) 2 α g s ( z , a ) d A ( z ) ) 1 2 + f g B p , α ( sup a D D | ϕ ( z ) | 2 ( 1 | ϕ ( z ) | p ) 2 α g s ( z , a ) d A ( z ) ) 1 2 + f g B p , α ( 1 α ) + | f ( 0 ) g ( 0 ) | p 2 C d ( f , g ; B p , α ) .

Thus C ϕ : B p , α Q ( p , s ) is Lipschitz continuous and the proof is completed. □

Remark 3.1 We know that a composition operator C ϕ : B p , α Q ( p , s ) is said to be bounded if there is a positive constant C such that C ϕ f Q ( p , s ) C f B p , α for all f B p , α . Theorem 3.1 shows that C ϕ : B p , α Q ( p , s ) is bounded if and only if it is Lipschitz-continuous, that is, if there exists a positive constant C such that
d ( f ϕ , g ϕ ; Q ( p , s ) ) C d ( f , g ; B p , α ) , for all  f , g B p , α .

By elementary functional analysis, a linear operator between normed spaces is bounded if and only if it is continuous, and the boundedness is trivially also equivalent to the Lipschitz-continuity. So, our result for composition operators in hyperbolic spaces is the correct and natural generalization of the linear operator theory.

Recall that a composition operator C ϕ : B p , α Q ( p , s ) is compact if it maps any ball in B p , α onto a precompact set in Q ( p , s ) .

The following observation is sometimes useful.

Proposition 3.1 Let 0 < p < , 0 s 1 and 0 < α 1 . Assume that ϕ is a holomorphic mapping from D into itself. If C ϕ : B p , α Q ( p , s ) is compact, it maps closed balls onto compact sets.

Proof If B B p , α is a closed ball and g Q ( p , s ) belongs to the closure of C ϕ ( B ) , we can find a sequence ( f n ) n = 1 B such that f n ϕ converges to g Q ( p , s ) as n . But ( f n ) n = 1 is a normal family, hence it has a subsequence ( f n j ) j = 1 converging uniformly on the compact subsets of D to an analytic function f. As in earlier arguments of Proposition 2.1 in [14], we get a positive estimate which shows that f must belong to the closed ball B. On the other hand, also the sequence ( f n j ϕ ) j = 1 converges uniformly on compact subsets to an analytic function, which is g Q ( p , s ) . We get g = f ϕ , i.e., g belongs to C ϕ ( B ) . Thus, this set is closed and also compact. □

Compactness of composition operators can be characterized in full analogy with the linear case.

Theorem 3.2 Let 0 < p < , 0 s 1 , and 0 < α 1 . Assume that ϕ is a holomorphic mapping from D into itself. Then the following statements are equivalent:
  1. (i)
    C ϕ : B p , α Q ( p , s ) is compact.(ii)
    lim r 1 sup a D | ϕ | r j | ϕ ( z ) | 2 ( 1 | ϕ ( z ) | p ) 2 α g s ( z , a ) d A ( z ) = 0 .
     

Proof We first assume that (ii) holds. Let B : = B ¯ ( g , δ ) B p , α , where g B p , α and δ > 0 , be a closed ball, and let ( f n ) n = 1 B be any sequence. We show that its image has a convergent subsequence in Q ( p , s ) , which proves the compactness of C ϕ by definition.

Again, ( f n ) n = 1 B ( D ) is a normal family, hence there is a subsequence ( f n j ) j = 1 which converges uniformly on the compact subsets of D to an analytic function f. By the Cauchy formula for the derivative of an analytic function, also the sequence ( f n j ) j = 1 converges uniformly to f . It follows that also the sequences ( f n j ϕ ) j = 1 and ( f n j ϕ ) j = 1 converge uniformly on the compact subsets of D to f ϕ and f ϕ , respectively. Moreover, f B B p , α since for any fixed R, 0 < R < 1 , the uniform convergence yields

Hence, d ( f , g ; B p , α ) δ .

Let ε > 0 . Since (ii) is satisfied, we may fix r, 0 < r < 1 , such that
sup a D | ϕ ( z ) | r | ϕ ( z ) | p 2 | ϕ ( z ) | 2 ( 1 | ϕ ( z ) | p ) 2 α g s ( z , a ) d A ( z ) ε .
By the uniform convergence, we may fix N 1 N such that
| f n j ϕ ( 0 ) f ϕ ( 0 ) | ε , for all  j N 1 .
(6)
The condition (ii) is known to imply the compactness of C ϕ : B p , α Q ( p , s ) , hence possibly to passing once more to a subsequence and adjusting the notations, we may assume that
f n j ϕ f ϕ Q ( p , s ) ε , for all  j N 2 ,  for some  N 2 N .
(7)
Now let
I 1 ( a , r ) = sup a D | ϕ ( z ) | r [ ( f n j ϕ ) p ( z ) ( f ϕ ) p ( z ) ] 2 g s ( z , a ) d A ( z ) ,
and
I 2 ( a , r ) = sup a D | ϕ ( z ) | r [ ( f n j ϕ ) p ( z ) ( f ϕ ) p ( z ) ] 2 g s ( z , a ) d A ( z ) .
Since ( f n j ) j = 1 B and f B , it follows that
I 1 ( a , r ) = sup a D | ϕ ( z ) | r [ ( f n j ϕ ) p ( z ) ( f ϕ ) p ( z ) ] 2 g s ( z , a ) d A ( z ) p 2 sup a D | ϕ ( z ) | r L ( f n j , f , ϕ ) g s ( z , a ) d A ( z ) d B α ( f n j , f ) sup a D | ϕ ( z ) | r | ϕ ( z ) | 2 ( 1 | ϕ ( z ) | p ) 2 α g s ( z , a ) d A ( z ) ,
where
L ( f n j , f , ϕ ) = | | ( f n j ϕ ) ( z ) | p 2 1 ( f n j ϕ ) ( z ) 1 | ( f n j ϕ ) ( z ) | p | ( f ϕ ) ( z ) | p 2 1 ( f ϕ ) ( z ) 1 | ( f ϕ ) ( z ) | p | 2 .
Hence,
I 1 ( a , r ) C ε .
(8)
On the other hand, by the uniform convergence on compact subsets of D , we can find an N 3 N such that for all j N 3 ,
L 1 ( f n j , f , ϕ ) = | | ( f n j ϕ ) ( z ) | p 2 1 f n j ( ϕ ( z ) ) 1 | f n j ( ϕ ( z ) ) | p | ( f ϕ ) ( z ) | p 2 1 f ( ϕ ( z ) ) 1 | f ( ϕ ( z ) ) | p | ε
for all z D with | ϕ ( z ) | r . Hence, for such j, we obtain
I 2 ( a , r ) = sup a D | ϕ ( z ) | r ( ( f n j ϕ ) p ( z ) ( f ϕ ) p ( z ) ) 2 g s ( z , a ) d A ( z ) sup a D | ϕ ( z ) | r L 1 ( f n j , f , ϕ ) | ϕ ( z ) | 2 g s ( z , a ) d A ( z ) ε ( sup a D | ϕ ( z ) | r | ϕ ( z ) | 2 ( 1 | ϕ ( z ) | p ) 2 α g s ( z , a ) d A ( z ) ) 1 2 C ε ,
hence,
I 2 ( a , r ) C ε ,
(9)

where C is the bound obtained from (iii) of Theorem 3.1. Combining (6), (7), (8) and (9), we deduce that f n j f in Q ( p , s ) .

As for the converse direction, let f n ( z ) : = 1 2 n α 1 z n for all n N , n 2 .
f B p , α = p 2 sup a D n α p 2 | z | α p 2 1 ( 1 | z | 2 ) α 1 2 p n p ( α 1 ) | z | n p ( 2 p 1 + 1 ) sup a D n α p 2 | z | α p 2 1 ( 1 | z | 2 ) α .
(10)
The function r n p 2 1 ( 1 r ) α attains its maximum at the point r = 1 α α + α p 2 1 . For simplicity, we see that (10) has the upper bound
( 2 p 1 + 1 ) n α ( 1 α α + n 1 ) n 1 ( α α + n 1 ) α ( 2 p 1 + 1 ) .

Then the sequence ( f n ) n = 1 belongs to the ball B ¯ ( 0 , ( 2 p 1 + 1 ) ) B p , α .

Suppose that C ϕ maps the closed ball B ¯ ( 0 , ( 2 p 1 + 1 ) ) B p , α into a compact subset of Q ( p , s ) ; hence, there exists an unbounded increasing subsequence ( n j ) j = 1 such that the image subsequence ( C ϕ f n j ) j = 1 converges with respect to the norm. Since both ( f n ) n = 1 and ( C ϕ f n j ) j = 1 converge to the zero function uniformly on compact subsets of D , the limit of the latter sequence must be zero. Hence,
n j α 1 ϕ n j Q ( p , s ) 0 , as  j .
(11)
Now let r j = 1 1 n j . For all numbers a, r j a < 1 , we have the estimate
n j α a n j 1 1 a n j 1 e ( 1 a ) α ( see [14] ) .
(12)
Using (12), we deduce
n j α 1 ϕ n j Q ( p , s ) 2 p 2 sup a D | ϕ | r j | n j α ( ϕ ( z ) ) n j 1 | ϕ n j ( z ) | p 2 1 ϕ ( z ) 1 | ϕ n j ( z ) | p | 2 g s ( z , a ) d A ( z ) C p 8 e 2 sup a D | ϕ | r j | ϕ ( z ) | 2 ( 1 | ϕ ( z ) | p ) 2 α g s ( z , a ) d A ( z ) .
(13)

From (11) and (13), the condition (ii) follows. This completes the proof. □

For 0 < p < and 0 s < , we define the weighted Dirichlet-class D ( p , s ) consists of those functions f H ( D ) for which
D | f ( z ) | p 2 | f ( z ) | 2 ( 1 | z | 2 ) s d A ( z ) < .
For 0 < p < and 0 s < , the generalized hyperbolic weighted Dirichlet-class D ( p , s ) consists of those functions f B ( D ) for which
D ( f p ( z ) ) 2 ( 1 | z | 2 ) s d A ( z ) < .

The proof of Proposition 2.2 implies the following corollary:

Corollary 3.1 For f , g D ( p , s ) . Then, D ( p , s ) is a complete metric space with respect to the metric defined by
d ( f , g ; D ( p , s ) ) : = d D ( p , s ) ( f , g ) + f g D ( p , s ) + | f ( 0 ) g ( 0 ) | p 2 ,
where
d D ( p , s ) ( f , g ) : = ( p 2 sup z D D | f ( z ) | f ( z ) | p 2 1 1 | f ( z ) | p g ( z ) | g ( z ) | p 2 1 1 | g ( z ) | p | 2 ( 1 | z | 2 ) s d A ( z ) ) 1 2 .

Moreover, the proofs of Theorems 3.1 and 3.2 yield the following result:

Theorem 3.3 Let 0 < p < , 1 < s 1 , and 0 < α 1 . Assume that ϕ is a holomorphic mapping from D into itself. Then the following statements are equivalent:
  1. (i)

    C ϕ : B p , α D ( p , s ) is Lipschitz continuous;

     
  2. (ii)
    C ϕ : B p , α D ( p , s ) is compact;(iii)
    D | ϕ ( z ) | 2 ( 1 | ϕ ( z ) | p ) 2 α ( 1 | z | 2 ) s d A ( z ) < .
     

Authors’ Affiliations

(1)
Faculty of Science, Mathematics Department, Sohag University, Sohag, 82524, Egypt
(2)
Faculty of Science, Mathematics Department, Taif University, Box 888, El-Hawiah, Taif, Saudi Arabia

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© El-Sayed Ahmed; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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