Stability of Admissible Functions

By using the concept of the weak subordination, we examine the stability a class of analytic functions in the unit disk is said to be stable if it is closed under weak subordination for a class of admissible functions in complex Banach spaces. The stability of analytic functions in the following classes is discussed: Bloch class, little Bloch class, hyperbolic little Bloch class, extend Bloch class Q p , and Hilbert Hardy class H 2 .


Introduction
We denote by U the unit disk {z ∈ C : |z| < 1} and by H U the space of all analytic functions in U. A function I, analytic in U, is said to be an inner function if and only if |I z | ≤ 1 such that |I e iθ | 1 almost everywhere.We recall that an inner function I can be factored in the form I BS where B is a Blaschke product and S is a singular inner function takes the form where μ is a finite positive Lebesgue measure.Let X and Y represent complex Banach spaces.The class of admissible functions G X, Y , consists of those functions f : X → Y that satisfy If f and g are analytic functions with f, g ∈ G X, Y , then f is said to be weakly subordinate to g, written as f≺ w g if there exist analytic functions φ, ω : U → X, with φ an inner function φ X ≤ 1 , so that f • φ g • ω.A class C of analytic functions in X is said to be stable if it is closed under weak subordination, that is, if f ∈ C whenever f and g are analytic functions in X with g ∈ C and f≺ w g.

International Journal of Mathematics and Mathematical Sciences
By making use of the above concept of the weak subordination, we examine the stability for a class of admissible functions in complex Banach spaces G X, Y .The stability of analytic functions appears in Bloch class, little Bloch class, hyperbolic little Bloch class, extend Bloch class Q p , and Hilbert Hardy class H 2 .

Stability of Bloch Classes
If f is an analytic function in U, then f is said to be a Bloch function if The space of all Bloch functions is denoted by B. The little Bloch space B 0 consists of those f ∈ B such that The hyperbolic Bloch class B h is defined by using the hyperbolic derivative in place of the ordinary derivative in the definition of the Bloch space, where the hyperbolic derivative of an analytic self-map ϕ : Similarly, we say ϕ ∈ B h 0 , the hyperbolic little Bloch class, if ϕ ∈ B h and lim Note that, for the function ϕ : U → X, we replace | • | by • X in the above definitions.
Theorem 2.1.Let X be a complex Banach space.If X contains all inner functions in U, then G X, X is stable.
Proof.Suppose that X contains all inner functions I : U → X.Take g z φ z z and ω z I z z ∈ U .Then, φ and ω are inner functions and I I • φ g • ω.Hence, I≺ w g, g ∈ X, and I ∈ X.Thus, X is stable.Theorem 2.2.Let X H U be a space of analytic functions in U which satisfies B ⊂ X.Then, G U, X is stable.
Proof.Suppose that X H U and B ⊂ X.Let f ∈ X, E {m ni : m, n ∈ Z} and F {z ∈ U : f z ∈ E}.Since F is a countable subset of U, it has capacity zero and therefore the universal covering map I from U onto U \ F is an inner function see, e.g., Chapter 2 of 1 .
hence I is an inner function in X.By putting g : f • I, we obtain that g ∈ X even though f ∈ X.Thus, X is stable and consequently yields the stability of G X, X .
Next we discuss the stability of the spaces B 0 and This operator is called the composition operator induced by ϕ.
Recall that a linear operator T : X → Y is said to be bounded if the image of a bounded set in X is a bounded subset of Y , while T is compact if it takes bounded sets to sets with compact closure.Furthermore, if T is a bounded linear operator, then it is called weakly compact, if T takes bounded sets in X to relatively weakly compact sets in Y .By using the operator C ϕ f, we have the following result.
Proof.Assume the analytic self-map ϕ of U and f ∈ B 0 ; thus, we have the linear operator C ϕ : B 0 → B 0 , defined by C ϕ f f • ϕ : g.By the assumption, we obtain that g is compact function in B 0 .Hence, ϕ is an inner function 4, Corollary 1.3 which implies the stability of G B 0 , B 0 .Theorem 2.5.Let ϕ be holomorphic self-map of U such that Then, G B 0 , B 0 is stable.
Proof.Assume the analytic self-map ϕ of U and f ∈ B 0 ; hence, in virtue of 5, Theorem 4.7 , it is implied that the composition operator C ϕ f on B 0 is compact.Thus we pose that G B 0 , B 0 is stable.
Theorem 2.6.Consider ϕ is a holomorphic self-map of U, satisfying the following condition: for every > 0, there exists 0 < r < 1 such that when |ϕ| > r.Then, G B 0 , B 0 is stable.
Proof.Assume the analytic self-map ϕ of U and f ∈ B 0 ; hence, in virtue of 5, Theorem 4.8 , it is yielded that the composition operator C ϕ f on B 0 is compact.Hence, we obtain that G B 0 , B 0 is stable.
Proof.According to 5, Theorem 4.10 , we have that C ϕ f is compact in B 0 and, consequently, G B 0 , B 0 is stable.
Theorem 2.8.Let ϕ be holomorphic self-map of U. If the function Proof.Assume the analytic self-map ϕ of U and f ∈ B h 0 .Since τ ϕ z is bounded, then in virtue of 4, Theorem 1.2 , it is yielded that ϕ is an inner function.By putting g : f • ϕ, where g ∈ B h 0 , we have the desired result.
Proof.Following 4 , it will be shown that there are inner functions ϕ ∈ B h 0 ; then, C ϕ : B 0 → B 0 is compact see 4, 5 .Thus, G B 0 , B 0 is stable.Theorem 2.10.Let ϕ be self-map in U and w : 0, 1 → 0, ∞ be continuous with lim t 0 w t 0 satisfying then G B 0 , B 0 is stable.
Proof.According to 5, Theorem 5.15 , we pose that ϕ is inner.Thus, in view of 4,

Stability of the Hilbert Hardy Space
In this section, we assume that f ∈ H Next we will show that the compactness of C ϕ f introduces the stability of G H 2 , H 2 .Two positive or complex measures μ and ν defined on a measurable space Ω, Σ are called singular if there exist two disjoint sets A and B in Σ whose union is Ω such that μ is zero on all measurable subsets of B while ν is zero on all measurable subsets of A. → H q is compact for all 0 < p < q < ∞ see 11, Theorem 6.4 .In view of Remark 3.4, we obtain that ϕ is an inner function; hence, G L p a , H q is stable.

Theorem 3.2. If the composition operator
where n ϕ w is the number of points in ϕ −1 w with multiplicity counted, then G H 2 , H 2 is stable.
Proof.According to 12, Corollary 3.6 , we have that C ϕ is compact on H 2 .Again in view of Remark 3.4, we obtain that ϕ is inner and, consequently, G H 2 , H 2 is stable.

Stability of Q p Class
For 0 < p < 1, an analytic function f in U is said to belong to the space where dA z dxdy rdrdθ is the Lebesgue area measure and g denotes the Green function for the disk given by The spaces Q p are conformally invariant.In 13 , It was shown that Q p B for all p, while Q 1 BMOA, the space of those f ∈ H 1 whose boundary values have bounded mean oscillation on ∂U see 14 .For 0 < α < 1, Λ α is the Lipschitz space, consisting of those f ∈ H U , which are continuous in U and satisfy for some C C f > 0. In this section, we will show the stability of functions belong to the spaces Q 1 and Λ α .
Proof.In the similar manner of Theorem 2.2, we pose an inner function ϕ on U. Now, in view of 15, Theorem H , yields g : for some z, then G Λ α , Λ α is stable.
Proof.Again as in Theorem 2.2, we obtain an inner function ϕ on U. Now in view of 15, Theorem K , yields g : f • ϕ ∈ Λ α , even though f ∈ Λ α .Thus, Λ α is stable.

Conclusion
From above, we conclude that the composition operator C ϕ , of admissible functions in different complex Banach spaces, plays an important role in stability of these spaces.It was shown that the compactness of this operator implied the stability, when ϕ is an inner function on the unit disk U. Furthermore, weakly compactness imposed the stability of Bloch spaces.
In addition, noncompactness leaded to the stability for some spaces such as Q p -spaces and Lipschitz spaces.

3 . 6 .
Next, we use the angular derivative criteria to discuss the stability of admissible functions.Recall that ϕ has angular derivative at ζ ∈ ∂U if the nontangential lim w f ζ ∈ ∂U exists and if f z − f ζ / z − ζ converges to some μ ∈ C as z → ζ nontangentially.International Journal of Mathematics and Mathematical Sciences Theorem If ϕ satisfies both the angular derivative criteria and . Moreover, Joel Shapiro obtained the following characterization of inner functions 6 : the function ϕ : U → U is inner if and only if Let ϕ be self-map of U and f ∈ H 2 .If 1.1 holds, then G H 2 , H 2 is stable.Proof.Assume the analytic self-map ϕ of U and f ∈ H 2 .Condition 1.1 implies that ϕ is an inner function.By setting g : C ϕ f f • ϕ and, consequently, g ∈ H 2 , it is yielded that G H 2 , H 2 is stable.
2, where H 2 is the Hilbert Hardy space on U, that is, the set of all analytic functions on U with square summable Taylor coefficients.It is well known that each such ϕ self-map in U induces a bounded composition operator C ϕ f f • ϕ onH 2 all the Aleksandrov measures of ϕ are singular absolutely continuous with respect to the arc-length measure see 7, 8 .Thus, in view of Remark 3.4.iIt is well known that if C ϕ is compact on H 2 , then it is compact on H p for all 0 < p < ∞ see 9, Theorem 6.1 .iiC ϕ is compact on H ∞ if and only if ϕ ∞ < 1 see 10, Theorem 2.8 .
4, Remark 1 , ϕ is inner.By letting g : C ϕ f f • ϕ, and, consequently, g ∈ H 2 , it is yielded that G H 2 , H 2 is stable.Theorem 3.3.If ϕ has values never approach the boundary of U, then G H 2 , H 2 is stable.Proof.Assume the composition operator C ϕ : H 2 → H 2 .Since ϕ has values never approach the boundary of U: ϕ ∞ sup ϕ , z ∈ U < 1, 3.2 C ϕ is compact on H 2 see 9, 10 .Hence, ϕ is an inner function and G H 2 , H 2 is stable.