Krzyz Conjecture The Theory And Methods

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.

"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"--

The volume is devoted to the interaction of modern scientific computation and classical function theory. Many problems in pure and more applied function theory can be tackled using modern computing facilities: numerically as well as in the sense of computer algebra. On the other hand, computer algorithms are often based on complex function theory, and dedicated research on their theoretical foundations can lead to great enhancements in performance. The contributions - original research articles, a survey and a collection of problems - cover a broad range of such problems.

Presents easy to understand proofs of same of the most difficult results about polynomials demonstrated by means of applications

This book gives a unified representation of generalizations of the Schwarz Lemma. It examines key coefficient theorems of the last century and explains the connection between coefficient estimates and characteristics of the hyperbolic geometry in a domain.