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## Über dieses Buch

​Inner functions form an important subclass of bounded analytic functions. Since they have unimodular boundary values, they appear in many extremal problems of complex analysis. They have been extensively studied since early last century, and the literature on this topic is vast. Therefore, this book is devoted to a concise study of derivatives of these objects, and confined to treating the integral means of derivatives and presenting a comprehensive list of results on Hardy and Bergman means. The goal is to provide rapid access to the frontiers of research in this field. This monograph will allow researchers to get acquainted with essentials on inner functions, and it is self-contained, which makes it accessible to graduate students.

## Inhaltsverzeichnis

### Chapter 1. Inner Functions

Abstract
The theory of Hardy spaces is a well established part of analytic function theory. Inner functions constitute a special family in this category. Therefore, it is natural to start with several topics on Hardy spaces and apply them in our discussions. However, we are not in a position to study this theory in detail and we assume that our readers have an elementary familiarity with this subject. In this chapter, we briefly mention, mostly without proof, the main theorems that we need in the study of inner functions. For a detailed study of this topic, we refer to [33].

### Chapter 2. The Exceptional Set of an Inner Function

Abstract
For a fixed $$w \in \mathbb{D}$$, the mapping
$${\tau }_{{}_{w}}(z) = \frac{w - z} {1 -\bar{ w}\,z},(z \in \mathbb{D}),$$
is an automorphism of the open unit disc $$\mathbb{D}$$.

### Chapter 3. The Derivative of Finite Blaschke Products

Abstract
Let $$w \in \mathbb{D}$$. Consider the Blaschke factor
$$b(z) = \frac{w - z} {1 -\bar{ w}\,z}.$$

### Chapter 4. Angular Derivative

Abstract
In this section, we find some elementary formulas for the derivative of B and S.

### Chapter 5. H p -Means of S′

Abstract
Let ϕ be an inner function, and let ϕ = BS be its canonical decomposition.

### Chapter 6. B p -Means of S′

Abstract
For an inner function ϕ, the H p -means of ϕ are not necessarily finite.

### Chapter 7. The Derivative of a Blaschke Product

Abstract
Let (z n ) n ≥ 1 be a Blaschke sequence and let
$$B(z) = \prod \limits _{n=1}^{\infty }\frac{\vert {z}_{n}\vert } {{z}_{n}} \,\, \frac{{z}_{n} - z} {1 -\bar{ {z}}_{n}\,z}.$$
For a fixed point $$z \in \mathbb{D}$$, we know that the partial products
$${B}_{N}(z) = \prod \limits _{n=1}^{N}\frac{\vert {z}_{n}\vert } {{z}_{n}} \,\, \frac{{z}_{n} - z} {1 -\bar{ {z}}_{n}\,z}$$
converge to B(z). Indeed, more is true.

### Chapter 8. H p -Means of B′

Abstract
Theorem 5.4 provided a necessary and sufficient condition for the inclusion $$B^{\prime} \in {H}^{p}(\mathbb{D})$$. This result, with some variation, is restated below. We will also see how the following lemma can be used to produce some families of Blaschke products that will be used throughout the text as illustrating examples.

### Chapter 9. B p -Means of B′

Abstract
According to Theorem 6.1, we have
$$B{\prime} \in {\bigcap \nolimits }_{0<p<\frac{1} {2} }{B}^{p}(\mathbb{D})$$
(9.1)
for any Blaschke product B. Compare this result with Theorem 7.12. There is a Blaschke product B such that $$B{\prime}\not\in {B}^{\frac{1} {2} }(\mathbb{D})$$. Hence, (9.1) is sharp and the Blaschke condition alone is not enough to conclude further results. Thus, we need to consider the Blaschke sequences which satisfy a stronger growth condition. Two such results are treated below.

### Chapter 10. The Growth of Integral Means of B′

Abstract
In all the preceding chapters, we studied various conditions under which some integral means of B′ were uniformly bounded. In this chapter, on the contrary, we assume that the integral means are not bounded and, in fact, they tend to infinity as a parameter varies. This parameter is usually the radius r which tends to 1. Our goal is to study the rate of growth of integral means as r→1.