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. Key Features • The book contains the recent achievements in trying to solve the Krzyż Conjecture (up to 2020) • It describes the chronological way the author had conducted his research in the math of complex analysis. In that respect the book is very different from standard mathematical texts in that it sketches the figure of a mathematics researcher as an actual human being and not just as a producer of new theorems and theories in mathematics • Hopefully that will reduce the "scary" image that many have on mathematicians

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