El-Sayed Ahmed Journal of Inequalities and Applications 2012, 2012:185 http://www.journalofinequalitiesandapplications.com/content/2012/1/185

RESEARCH

Open Access

Natural metrics and composition operators in generalized hyperbolic function spaces A El-Sayed Ahmed* *

Correspondence: [email protected] Faculty of Science, Mathematics Department, Sohag University, Sohag, 82524, Egypt Current address: Faculty of Science, Mathematics Department, Taif University, Box 888, El-Hawiah, Taif, Saudi Arabia

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 Cφ 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| < } 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)| <  for all z ∈ D. Also, dA(z) be the normalized area measure on D so that A(D) ≡ . The usual α-Bloch spaces Bα and Bα, are defined as the sets of those f ∈ H(D) for which   α f Bα = supf  (z)  – |z| < ∞, z∈D

and   α lim f  (z)  – |z| = ,

|z|→

respectively. Now, we will give the following definition: Definition . The (p, α)-Bloch spaces Bp,α and Bp,α, are defined as the sets of those f ∈ H(D) for which f Bp,α =

  p –   α p supf (z)  f  (z)  – |z| < ∞,  z∈D

and   p –   α lim f (z)  f  (z)  – |z| = ,

|z|→

where  < p, α < ∞. © 2012 El-Sayed Ahmed; licensee Springer. 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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Remark . 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 =  in Definition ., we will obtain the spaces Bα and Bα, . Remark . (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 []). If (X, d) is a metric space, we denote the open and closed balls with center x and radius r > ¯ r) := {y ∈ X : d(y, x) = r}, respectively. The well by B(x, r) := {y ∈ X : d(y, x) < r} and B(x, |f  (z)| known hyperbolic derivative is defined by f * (z) = –|f of f ∈ B(D) and the hyperbolic (z)| +|f (z)| ) between f (z) and zero. distance is given by ρ(f (z), ) :=  log( –|f (z)| A function f ∈ B(D) is said to belong to the hyperbolic α-Bloch class Bα* if

α  f Bα* = sup f * (z)  – |z| < ∞. z∈D

* The little hyperbolic Bloch-type class Bα, consists of all f ∈ Bα* such that

 α lim f * (z)  – |z| = .

|z|→

The Schwarz-Pick lemma implies Bα* = B(D) for all α ≥  with f Bα* ≤ , and therefore, the hyperbolic α-Bloch classes are of interest only when  < α < . It is obvious that Bα* is not a linear space since the sum of two functions in B(D) does not necessarily belong to B(D). p  – 

|f (z)| of f ∈ Now, let  < p < ∞, we define the hyperbolic derivative by fp* (z) = p |f (z)| –|f (z)|p B(D). When p = , 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 * Bp,α if

  *  α f Bp,α < ∞. * = sup fp (z)  – |z| z∈D

* * consists of all f ∈ Bp,α such The little generalized (p, α)-hyperbolic Bloch-type class Bp,α, that

α  lim fp* (z)  – |z| = .

|z|→

a–z Let the Green’s function of D be defined as g(z, a) = log |ϕa(z)| , where ϕa (z) = – is the ¯ az Möbius transformation related to the point a ∈ D. For  < p, s < ∞, the hyperbolic class Q* (p, s) consists of those functions f ∈ B(D) for which p f Q* (p,s)

 = sup a∈D

D

 *  s fp (z) g (z, a) dA(z) < ∞.

Moreover, we say that f ∈ Q* (p, s) belongs to the class Q* (p, s, ) if  lim

|a|→ D



 fp* (z) g s (z, a) dA(z) = .

When p = , we obtain the usual hyperbolic Q class as studied in [, , ].

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

For any holomorphic self-mapping φ of D, the symbol φ induces a linear composition operator Cφ (f ) = f ◦ φ from H(D) or B(D) into itself. The study of a composition operator Cφ acting on the spaces of analytic functions has engaged many analysts for many years (see, e.g., [–, , , ] and others). Yamashita was probably the first to consider systematically hyperbolic function classes. He introduced and studied hyperbolic Hardy, BMOA and Dirichlet classes in [–] and others. More recently, Smith studied inner functions in the hyperbolic little Bloch-class [], and the hyperbolic counterparts of the Qp spaces were studied by Li in [] and Li et al. in []. Further, hyperbolic Qp classes and composition operators were studied by Pérez-González et al. in []. * and the In this paper, we will study the generalized hyperbolic (p, α)-Bloch classes Bp,α * 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 * to Q* (p, s) classes by conditions depending on the symbol φ only. (p, α)-Bloch class Bp,α Thus, the results are generalizations of the recent results of Pérez-González, Rättyä and Taskinen []. Recall that a linear operator T : X → Y is said to be bounded if there exists a constant C >  such that T(f )Y ≤ Cf 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 (xn ) ∈ X, the sequence (Txn ) ∈ 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 []. Hence we omit the proof. Lemma . Assume φ is a holomorphic mapping from D into itself. Let  < p, s < ∞, and * → Q* (p, s) is compact if and only if for any bounded sequence  < α < ∞. Then Cφ : Bp,α * which converges to zero uniformly on compact subsets of D as n → ∞, we (fn )n∈N ∈ Bp,α have limn→∞ Cφ fn Q* (p,s) = . Using the standard arguments similar to those outlined in Lemma  of [], we have the following lemma: * Lemma . Let  < α < ∞, then there exist two functions f , g ∈ Bp,α such that for some constant C,

  *   *  f (z) + g (z)  – |z| α ≥ C > , p p

for each z ∈ D.

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* * 2 Natural metrics in Bp, α and Q (p, s) classes In this section we introduce natural metrics on generalized hyperbolic α-Bloch classes * Bp,α and the classes Q* (p, s). * (see []) by Let  < p, s < ∞, and  < α < . First, we can find a natural metric in Bp,α defining

 p   *  := dBp,α d f , g; Bp,α * (f , g) + f – gBp,α + f () – g() ,

()

where    p p  f (z)|f (z)|  – g  (z)|g(z)|  –     – |z| α . – dBp,α * (f , g) := sup   – |f (z)|p  p  – |g(z)| z∈D For f , g ∈ Q* (p, s), define their distance by  p   d f , g; Q* (p, s) := dQ* (f , g) + f – gQ(p,s) + f () – g()  , where  dQ* (f , g) :=

p sup  z∈D

 p p       f (z)|f (z)|  – g  (z)|g(z)|  –  s   g (z, a) dA(z) . –   – |f (z)|p  – |g(z)|p  D

* ). Now, we give a characterization of the complete metric space d(·, ·; Bp,α * * Proposition . The class Bp,α equipped with the metric d(·, ·; Bp,α ) is a complete metric * * is a closed (and therefore complete) subspace of Bp,α . space. Moreover, Bp,α, * * * Proof Clearly d(f , g; Bp,α ) ≥ , d(f , g, Bp,α ) = d(g, f ; Bp,α ). Also,

      * * * ≤ d f , g; Bp,α + d g, h; Bp,α . d f , h; Bp,α * * Moreover, d(f , f ; Bp,α ) =  for all f , g, h ∈ Bp,α . * ) =  implies It follows from the presence of the usual (p, α)-Bloch term that d(f , g; Bp,α ∞ * f = g. Hence, (Bp,α , d) is a metric space. Let (fn )n= be a Cauchy sequence in the metric * , d), that is, for any ε > , there is an N = N(ε) ∈ N such that space (Bp,α

  * d fn , fm ; Bp,α <ε for all n, m > N . Since (fn ) ⊂ B(D), the family (fn ) is uniformly bounded and hence normal in D. Therefore, there exist f ∈ B(D) and a subsequence (fnj )∞ j= such that fnj 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    p p  f (z)|f (z)|  – fm (z)|fm (z)|  –    α     – |f (z)|p –  – |f (z)|p   – |z| m    p p  f (z)|fn (z)|  – fm (z)|fm (z)|  –       – |z| α ≤ lim d fn , fm ; B * ≤ ε = lim  n – p,α  p p n→∞ n→∞  – |fn (z)|  – |fm (z)|

()

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for all z ∈ D, and it follows that f Bp,α * ≤ fm B * + ε. p,α * Thus, f ∈ Bp,α as desired. Moreover, () and the completeness of the usual (p, α)-Bloch im∞ ply that (fn )n= converges to f with respect to the metric d. The second part of the assertion follows by (). 

Next, we give a characterization of the complete metric space d(·, ·; Q* (p, s)). Proposition . The class Q* (p, s) equipped with the metric d(·, ·; Q* (p, s)) is a complete metric space. Moreover, Q* (p, s, ) 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)) ≥ , • d(f , f ; Q* (p, s)) = , • d(f , g; Q* (p, s)) =  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 (fn )∞ n= be a Cauchy sequence in the metric space * (Q (p, s), d), that is, for any ε >  there is an N = N(ε) ∈ N such that d(fn , fm ; Q* (p, s)) < ε, for all n, m > N . Since fn ∈ B(D) such that fn converges to f uniformly on compact subsets of D. Let m > N and  < r < . Then Fatou’s lemma yields   p p  |f (z)|  – f  (z) |fm (z)|  – fm (z)  s     – |f (z)|p –  – |f (z)|p  g (z, a) dA(z) m D(,r)   p p   |fn (z)|  – fn (z) |fm (z)|  – fm (z)  s  g (z, a) dA(z) = lim  –  – |fn (z)|p  – |fm (z)|p  D(,r) n→∞  p p    |fn (z)|  – fn (z) |fm (z)|  – fm (z)  s   g (z, a) dA(z) ≤ ε , – ≤ lim  n→∞ D  – |fn (z)|p  – |fm (z)|p 



and by letting r → – , it follows that  D



 fp* (z) g s (z, a) dA(z) ≤ ε

 p    |fm (z)|  – fm (z)  s   g (z, a) dA(z). +   – |fm (z)|p  D

()

This yields p

f Q* (p,s) ≤ ε + fm Q* (p,s) , and thus f ∈ Q* (p, s). We also find that fn → f with respect to the metric of Q* (p, s). The second part of the assertion follows by (). 

3 Lipschitz continuous and compactness of Cφ Theorem . Let  < p < ∞,  ≤ s ≤ , and  < α ≤ . Assume that φ is a holomorphic mapping from D into itself. Then the following statements are equivalent:

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* (i) Cφ : Bp,α → Q* (p, s) is bounded; * (ii) Cφ : Bp,α → Q* (p, s) is Lipschitz continuous;  |φ  (z)| s (iii) supa∈D D (–|φ(z)| p )α g (z, a) dA(z) < ∞.

Proof First, assume that (i) holds, then there exists a constant C such that Cφ f Q* (p,s) ≤ Cf Bp,α * ,

* for all f ∈ Bp,α .

* * , the function ft (z) = f (tz), where  < t < , belongs to Bp,α with the propFor given f ∈ Bp,α erty ft Bp,α * ≤ f B * . Let f , g be the functions from Lemma . such that p,α

     ≤ fp* (z) + gp* (z),  α ( – |z| ) for all z ∈ D, so that |φ  (z)| ≤ (f ◦ φ)* (z) + (g ◦ φ)* (z). ( – |φ(z)|)α Thus, 

|tφ  (z)| g s (z, a) dA(z)  α D ( – |tφ(z)| )       ≤C (f ◦ tφ)*p (z) + (g ◦ tφ)*p (z) g s (z, a) dA(z) D

  ≤ CCφ  f B* + gB* . p,α

p,α

This estimate together with the Fatou’s lemma implies (iii). * , we see that Conversely, assuming that (iii) holds and that f ∈ Bp,α  sup a∈D

D

  (f ◦ φ)*p (z) g s (z, a) dA(z)

= sup a∈D

 D

  *   fp φ(z) φ  (z) g s (z, a) dA(z) 

≤ f B* sup p,α

a∈D

D

|φ  (z)| g s (z, a) dA(z). ( – |φ(z)|p )α

Hence, it follows that (i) holds. * → Q* (p, s) is Lipschitz continuous, that is, there (ii) ⇐⇒ (iii). Assume first that Cφ : Bp,α exists a positive constant C such that     * , d f ◦ φ, g ◦ φ; Q* (p, s) ≤ Cd f , g; Bp,α

* for all f , g ∈ Bp,α .

Taking g = , this implies  p    , f ◦ φQ* (p,s) ≤ C f Bp,α * + f Bp,α + f ()

* for all f ∈ Bp,α .

()

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The assertion (iii) for α =  follows by choosing f (z) = z in (). If  < α < , then    p  z  p  p    f (z)  =  f (t)  f  (t) dt + f ()   p   |z|  p dx + f ()  ≤ f Bp,α  α ( – x )  p f Bα  + f ()  , ≤ ( – α) this yields p    p f – gBp,α     f φ() – g φ()   ≤ + f () – g()  . p ( – α) * such that Moreover, Lemma . implies the existence of f , g ∈ Bp,α

  *  f (z) + g * (z)  – |z| α ≥ C > , p p

for all z ∈ D.

()

Combining () and (), we obtain  p  p   f Bp,α * + gB * + f Bp,α + gBp,α + f () + g() p,α  |φ  (z)| ≥C g s (z, a) dA(z) p α D ( – |φ(z)| ) for which the assertion (iii) follows. Assume now that (iii) is satisfied, we have   d f ◦ φ, g ◦ φ; Q* (p, s) = dQ* (f ◦ φ, g ◦ φ) + f ◦ φ – g ◦ φQ(p,s) (p,s)

     p + f φ() – g φ()     ≤ dBp,α * (f , g) sup

  |φ  (z)| s g (z, a) dA(z) p α a∈D D ( – |φ(z)| )     |φ  (z)| s + f – gBp,α sup g (z, a) dA(z) p α a∈D D ( – |φ(z)| ) +

f – gBp,α

( – α)   * . ≤ Cd f , g; Bp,α

 p + f () – g() 

* Thus Cφ : Bp,α → Q* (p, s) is Lipschitz continuous and the proof is completed.



* → Q* (p, s) is said to be Remark . We know that a composition operator Cφ : Bp,α * bounded if there is a positive constant C such that Cφ f Q* (p,s) ≤ Cf Bp,α for all f ∈ Bp,α . * * Theorem . shows that Cφ : Bp,α → Q* (p, s) is bounded if and only if it is Lipschitzcontinuous, that is, if there exists a positive constant C such that

    * , d f ◦ φ, g ◦ φ; Q* (p, s) ≤ Cd f , g; Bp,α

* for all f , g ∈ Bp,α .

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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. * → Q* (p, s) is compact if it maps any ball in Recall that a composition operator Cφ : Bp,α * Bp,α onto a precompact set in Q* (p, s). The following observation is sometimes useful.

Proposition . Let  < p < ∞,  ≤ s ≤  and  < α ≤ . Assume that φ is a holomorphic * → Q* (p, s) is compact, it maps closed balls onto mapping from D into itself. If Cφ : Bp,α compact sets. * Proof If B ⊂ Bp,α is a closed ball and g ∈ Q* (p, s) belongs to the closure of Cφ (B), we can ∞ * find a sequence (fn )∞ n= ⊂ B such that fn ◦ φ converges to g ∈ Q (p, s) as n → ∞. But (fn )n= is a normal family, hence it has a subsequence (fnj )∞ j= converging uniformly on the compact subsets of D to an analytic function f . As in earlier arguments of Proposition . in [], we get a positive estimate which shows that f must belong to the closed ball B. On the other hand, also the sequence (fnj ◦ φ)∞ j= 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 Cφ (B). Thus, this set is closed and also compact. 

Compactness of composition operators can be characterized in full analogy with the linear case. Theorem . Let  < p < ∞,  ≤ s ≤ , and  < α ≤ . Assume that φ is a holomorphic mapping from D into itself. Then the following statements are equivalent: * (i) Cφ : Bp,α → Q* (p, s) is compact. (ii)  lim– sup

r→ a∈D

|φ|≥rj

|φ  (z)| g s (z, a) dA(z) = . ( – |φ(z)|p )α

* * ¯ δ) ⊂ Bp,α , where g ∈ Bp,α and δ > , be a Proof We first assume that (ii) holds. Let B := B(g, ∞ closed ball, and let (fn )n= ⊂ B be any sequence. We show that its image has a convergent subsequence in Q* (p, s), which proves the compactness of Cφ by definition. ∞ Again, (fn )∞ n= ⊂ B(D) is a normal family, hence there is a subsequence (fnj )j= 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 (fnj )∞ j= converges uni∞ ∞   formly to f . It follows that also the sequences (fnj ◦ φ)j= and (fnj ◦ φ)j= converge uniformly * on the compact subsets of D to f ◦ φ and f  ◦ φ, respectively. Moreover, f ∈ B ⊂ Bp,α since for any fixed R,  < R < , the uniform convergence yields

   p p  f (z)|f (z)|  – g  (z)|g(z)|  –     – |z| α – sup  p p   – |f (z)|  – |g(z)| |z|≤R   p –    p – α  + sup f (z) – g(z)  f  (z) – g  (z)  – |z| + f () – g()  |z|≤R

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 f  (z)|f (z)| p –  p  nj nj α g  (z)|g(z)|  –   = lim sup  –  – |z|  p p j→∞ |z|≤R  – |fnj (z)|  – |g(z)|   p –    p – α  + sup fnj (z) – g(z)  fnj (z) – g  (z)  – |z| + f () – g()  ≤ δ. |z|≤R

* Hence, d(f , g; Bp,α ) ≤ δ. Let ε > . Since (ii) is satisfied, we may fix r,  < r < , such that

 sup a∈D

|φ(z)|≥r

|φ(z)|p– |φ  (z)| s g (z, a) dA(z) ≤ ε. ( – |φ(z)|p )α

By the uniform convergence, we may fix N ∈ N such that   fn ◦ φ() – f ◦ φ() ≤ ε, j

for all j ≥ N .

()

The condition (ii) is known to imply the compactness of Cφ : Bp,α → Q(p, s), hence possibly to passing once more to a subsequence and adjusting the notations, we may assume that fnj ◦ φ – f ◦ φQ(p,s) ≤ ε,

for all j ≥ N , for some N ∈ N.

()

Now let  I (a, r) = sup



|φ(z)|≥r

a∈D

 (fnj ◦ φ)*p (z) – (f ◦ φ)*p (z) g s (z, a) dA(z),

and  I (a, r) = sup a∈D

|φ(z)|≤r



 (fnj ◦ φ)*p (z) – (f ◦ φ)*p (z) g s (z, a) dA(z).

Since (fnj )∞ j= ⊂ B and f ∈ B, it follows that  I (a, r) = sup a∈D

|φ(z)|≥r

p ≤ sup  a∈D

 (fnj ◦ φ)*p (z) – (f ◦ φ)*p (z) g s (z, a) dA(z)



|φ(z)|≥r

≤ dBα* (fnj , f ) sup a∈D

L(fnj , f , φ)g s (z, a) dA(z)  |φ(z)|≥r

|φ  (z)| g s (z, a) dA(z), ( – |φ(z)|p )α

where p   p  |(fnj ◦ φ)(z)|  – (fnj ◦ φ) (z) |(f ◦ φ)(z)|  – (f ◦ φ) (z)   . L(fnj , f , φ) =  –   – |(fnj ◦ φ)(z)|p  – |(f ◦ φ)(z)|p

Hence, I (a, r) ≤ Cε.

()

El-Sayed Ahmed Journal of Inequalities and Applications 2012, 2012:185 http://www.journalofinequalitiesandapplications.com/content/2012/1/185

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On the other hand, by the uniform convergence on compact subsets of D, we can find an N ∈ N such that for all j ≥ N ,  |(f ◦ φ)(z)| p – f  (φ(z))  p  nj |(f ◦ φ)(z)|  – f  (φ(z))  nj L (fnj , f , φ) =  – ≤ε  – |fnj (φ(z))|p  – |f (φ(z))|p for all z ∈ D with |φ(z)| ≤ r. Hence, for such j, we obtain  I (a, r) = sup a∈D

≤ sup a∈D



|φ(z)|≤r



|φ(z)|≤r

  ≤ ε sup a∈D

 (fnj ◦ φ)*p (z) – (f ◦ φ)*p (z) g s (z, a) dA(z)

  L (fnj , f , φ)φ  (z) g s (z, a) dA(z)

|φ(z)|≤r

|φ  (z)| g s (z, a) dA(z) ( – |φ(z)|p )α

 

≤ Cε,

hence, I (a, r) ≤ Cε,

()

where C is the bound obtained from (iii) of Theorem .. Combining (), (), () and (), we deduce that fnj → f in Q* (p, s). As for the converse direction, let fn (z) :=  nα– zn for all n ∈ N, n ≥ . αp

αp

n  |z|  – ( – |z| )α p sup  a∈D  – –p np(α–) |z|np α   αp αp  ≤ p– +  sup n  |z|  –  – |z| .

= f Bp,α *

()

a∈D np

The function r  – ( – r)α attains its maximum at the point r =  – we see that () has the upper bound 

  p– +  nα  –

α α+n–

n– 

α α+n–

α

α . αp α+  –

For simplicity,

  ≤ p– +  .

p– * ¯ + )) ⊂ Bp,α . Then the sequence (fn )∞ n= belongs to the ball B(, ( p– * ¯ Suppose that Cφ maps the closed ball B(, ( + )) ⊂ Bp,α into a compact subset of Q* (p, s); hence, there exists an unbounded increasing subsequence (nj )∞ j= such that the ∞ image subsequence (Cφ fnj )j= converges with respect to the norm. Since both (fn )∞ n= and ∞ (Cφ fnj )j= converge to the zero function uniformly on compact subsets of D, the limit of the latter sequence must be zero. Hence,

 α– n  n φ j  * → , j Q (p,s) Now let rj =  – nαj anj – –a

nj

≥

 . nj

as j → ∞.

()

For all numbers a, rj ≤ a < , we have the estimate

 e( – a)α



 see [] .

()

El-Sayed Ahmed Journal of Inequalities and Applications 2012, 2012:185 http://www.journalofinequalitiesandapplications.com/content/2012/1/185

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Using (), we deduce  α– n  p n φ j  * ≥ sup j Q (p,s)  a∈D ≥

 |φ|≥rj

Cp sup e a∈D

p  α   nj (φ(z))nj – |φ nj (z)|  – φ  (z)  s   g (z, a) dA(z)    – |φ nj (z)|p



|φ|≥rj

|φ  (z)| g s (z, a) dA(z). ( – |φ(z)|p )α

From () and (), the condition (ii) follows. This completes the proof.

() 

For  < p < ∞ and  ≤ s < ∞, we define the weighted Dirichlet-class D(p, s) consists of those functions f ∈ H(D) for which  D

      f (z)p– f  (z)  – |z| s dA(z) < ∞.

For  < p < ∞ and  ≤ s < ∞, the generalized hyperbolic weighted Dirichlet-class D* (p, s) consists of those functions f ∈ B(D) for which  D



fp* (z)

 

s  – |z| dA(z) < ∞.

The proof of Proposition . implies the following corollary: Corollary . For f , g ∈ D* (p, s). Then, D* (p, s) is a complete metric space with respect to the metric defined by  p   d f , g; D* (p, s) := dD* (p,s) (f , g) + f – gD(p,s) + f () – g()  , where  dD* (p,s) (f , g) :=

p sup  z∈D

 p p       f (z)|f (z)|  – g  (z)|g(z)|  –     s     – |f (z)|p –  – |g(z)|p   – |z| dA(z) . D

Moreover, the proofs of Theorems . and . yield the following result: Theorem . Let  < p < ∞, – < s ≤ , and  < α ≤ . Assume that φ is a holomorphic mapping from D into itself. Then the following statements are equivalent: * (i) Cφ : Bp,α → D* (p, s) is Lipschitz continuous; * → D* (p, s) is compact; (ii) Cφ : Bp,α (iii)  D

 s |φ  (z)|  – |z| dA(z) < ∞. ( – |φ(z)|p )α

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

El-Sayed Ahmed Journal of Inequalities and Applications 2012, 2012:185 http://www.journalofinequalitiesandapplications.com/content/2012/1/185

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doi:10.1186/1029-242X-2012-185 Cite this article as: El-Sayed Ahmed: Natural metrics and composition operators in generalized hyperbolic function spaces. Journal of Inequalities and Applications 2012 2012:185.

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