We study boundedness and compactness of composition operators in the generalized Hölder-type space of holomorphic functions in the unit disc with prescribed modulus of continuity. We also devote a significant part of the article to outline some embeddings between such Hölder-type spaces, to discuss properties of modulus of continuity and to construct some useful examples. KEYWORDS composition operators, Hölder spaces, holomorphic functions, modulus of continuity MSC CLASSIFICATION 47B33; 46E15; 30H99 1 INTRODUCTION Let D stand for the unit disc in complex plane C. Given a holomorphic function ∶ D → D by C , we denote the linear composition operator C f(z) ∶= f• (z) = f((z)). The study of the boundedness and compactness of the composition operator C in various function spaces over the unit disc and more general domains is a classical problem studied by many researchers. Without claiming for completeness, we mention the following spaces and references: the Besov space, 1-4
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