In the present work we develop a unified way of looking at normed spaces of analytic functions (NSAF's) and their embedding into the Frechet space of analytic functions on a general domain, by requiring only that the embedding map is bounded. This is a succinct definition of NSAF and derive from it a list of interesting properties. For example Proposition 4.4 describes the behavior of point evaluations and Proposition 4.6 part (i) gives a general sufficient condition for a NSAF to be a Banach space, which as far as we know, are new results. Also, Proposition 4.5, parts (ii) and (iii) of Proposition 4.6 and Proposition 4.7 are results, which are slight generalizations of fairly standard results, which show up elsewhere in a more specific setting. Some of the facts about NSAF's are stated and proven in a more general context. In particular, a significant part of the material is dedicated to the normed space of continuous functions on a metric space. On the other hand, we provide the necessary background on differential geometry and complex analysis, which further determine the peculiarities in the context of spaces of analytic functions. At the end we illustrate our results on two specific examples of NSAF's, namely the Bergman and the Bloch Spaces over a general domain in Cd. We give a new proof of the reflexivity of the Bergman Space Ap(G, μ) for the case p>1 and of the Schur property of A1(G, μ). We also give new proofs for the equivalences of some of the definitions of the Bloch functions.

In the present work we develop a unified way of looking at normed spaces of analytic functions (NSAF's) and their embedding into the Frechet space of analytic functions on a general domain, by requiring only that the embedding map is bounded. This is a succinct definition of NSAF and derive from it a list of interesting properties. For example Proposition 4.4 describes the behavior of point evaluations and Proposition 4.6 part (i) gives a general sufficient condition for a NSAF to be a Banach space, which as far as we know, are new results. Also, Proposition 4.5, parts (ii) and (iii) of Proposition 4.6 and Proposition 4.7 are results, which are slight generalizations of fairly standard results, which show up elsewhere in a more specific setting. Some of the facts about NSAF's are stated and proven in a more general context. In particular, a significant part of the material is dedicated to the normed space of continuous functions on a metric space. On the other hand, we provide the necessary background on differential geometry and complex analysis, which further determine the peculiarities in the context of spaces of analytic functions. At the end we illustrate our results on two specific examples of NSAF's, namely the Bergman and the Bloch Spaces over a general domain in Cd. We give a new proof of the reflexivity of the Bergman Space Ap(G, μ) for the case p>1 and of the Schur property of A1(G, μ). We also give new proofs for the equivalences of some of the definitions of the Bloch functions.

In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.

In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.