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Семинар Геометрија, визуелизација и образовање са применама

Математички Институт САНУ, Кнез Михајлова 35, сала 301ф


Четвртак, 15. новембар 2018. у 17:15 часова

Предавач: Nikola Tuneski, Cyril and Methodius University in Skopje, Republic of Macedonia

UNIVALENT FUNCTIONS: SHORT INTRODUCTION AND SOME NEW RESULTS

Апстракт: A complex function of one variable is said to be univalent if it is one-on-one and onto. The Riemann mapping theorem gives right to consider only univalent functions defined on the unit disk, while a result of Titchmarsh justifies normalisation of univalent functions in a way that f(0) = f'(0) - 1 = 0, i.e., f(z) = z + a2z2 + a3z3 + ..... These functions form the class of univalent functions S. The Bieberbach conjecture from 1916 busted the interest for the class S which resulted in development of variety of new methods and delivery of plenty significant results. It was finally proven by de Branges in 1984. Never the less, the interest for univalent functions remaind. The class S is large and the study is usually focused on its subclasses that carry quite descriptive names such as starlike functions, convex functions, close-to-convex functions, functions with bounded turning etc. There are two major directions for research: find sufficient condition that embed certain function in the class of univalent functions or some of its subclasses (the condition is usually over a simple expression involving f, f', f'' or over the coefficients from the expansion of f); study the geometrical and analytical properties of functions in S or its subclasses. The lecture will give definitions and explanations of the main terms, as well as explanation of the methods that I most frequently use in my research (differential subordinations and Clunie-Jack lemma). The focus will be on presentation of selected results from the theory with highlights of mine contribution.





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