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This book is about one of the beautiful topics in mathematics. It describes an on-going research on bounded analytic functions which are defined on the unit disc. This is a very active topic that belongs to the theory of complex analysis in a single complex variable. Complex analysis is one of the classical chapters in Mathematics. It contains the analytic theory of functions, the geometric function theory among other theoretical areas, as well as many applications. Some applications originate in other fields of mathematics: geometry, topology, arithmetic and number theory in general, algebra etc. Other applications originate in other scientific and engineering disciplines: physics, dynamical systems, electrical engineering etc.The book includes much more than just a review on the Krzyż conjecture. It includes topics on inner functions within the context of problems that are different from the Krzyż conjecture as well as other topics on general bounded analytic functions. Progress in mathematical research is frequently fuelled by efforts to solve open problems. The book also includes a few important open problems and some partial solutions of these.

Bernstein functions appear in various fields of mathematics, e.g. probability theory, potential theory, operator theory, functional analysis and complex analysis – often with different definitions and under different names. Among the synonyms are `Laplace exponent'instead of Bernstein function, and complete Bernstein functions are sometimes called `Pick functions', `Nevanlinna functions'or `operator monotone functions'. This monograph – now in its second revised and extended edition – offers a self-contained and unified approach to Bernstein functions and closely related function classes, bringing together old and establishing new connections. For the second edition the authors added a substantial amount of new material. As in the first edition Chapters 1 to 11 contain general material which should be accessible to non-specialists, while the later Chapters 12 to 15 are devoted to more specialized topics. An extensive list of complete Bernstein functions with their representations is provided.

This text is a self-contained and unified approach to Bernstein functions and their subclasses, bringing together old and establishing new connections. Applications of Bernstein functions in different fields of mathematics are given, with special attention to interpretations in probability theory. An extensive list of complete Bernstein functions with their representations is provided.

This work is a fresh presentation of the Ahlfors-Weyl theory of holomorphic curves that takes into account some recent developments in Nevanlinna theory and several complex variables. The treatment is differential geometric throughout, and assumes no previous acquaintance with the classical theory of Nevanlinna. The main emphasis is on holomorphic curves defined over Riemann surfaces, which admit a harmonic exhaustion, and the main theorems of the subject are proved for such surfaces. The author discusses several directions for further research.