"Babes-Bolyai" University of Cluj-Napoca
Faculty of Mathematics and Computer Science
| Geometric function theory |
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| Teaching Staff in Charge |
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| Aims |
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The presentation of principal classes of univalent functions defined by remarkable geometric properties and some of their applications in the theory of conformal mappings. |
| Content |
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1. Univalent functions; classical results. Aria's Theorem. Covering Theorem for the S class (Koebe, Bieberbach). Covering Theorem for the Sigma class. Distortion Theorems (Koebe, Bieberbach). The compactness of the S class. The Bieberbach's conjecture.
2. Analytical functions with real positive part. Subordination. - Integral representation; Herglotz's formula. The theorems of Herglotz. - Representations by Stiltjes integrals. Caratheodory's Theorem. - Bounds of holomorphic functions with real positive part. - Subordination; the subordination principle (Lindelof). Sakaguchi's Lemma. 3. Special classes of univalent functions. - Starlike functions. Radius of starlikeness. Theorem about the coefficient bounds of functions from S^*. Structure formula. - Convex functions. Duality's Theorem (Alexander). The compactness of K class. Radius of convexity. - Alpha - convex functions. The Theorem of starlikeness of alpha - convex functions. Duality's Theorem. Radius of alpha - convexity. Bounds Theorems (Miller). - Close - to - convex functions. Univalence criteria of Noshiro - Warschawski - Wolff. Univalence criteria of Ozaki - Kaplan. Characterizing Theorem of close - to - convex functions (Kaplan). Linear accessible domains. - Typical real functions. Structure formula. Duality Theorem. Theorem about the coefficients. A sufficient condition for the univalence of the typical real functions. Consequence (Aksentiev). Thalk - Chakalov Theorem. Univalence criteria for meromorphic functions. Aksentiev's Theorem. Starlikeness and convexity conditions for meromorphic functions. 4. Diffeomorphism conditions in the complex plane. - Spirallike generalized functions of C^1 class. General theorems; particular cases. - Nonanalytic alpha - convex function. Preliminary lemmas. The Theorem of starlikeness of alpha - convex nonanalytic functions. Examples. - C^1 transforms and the refraction law. - Close- to -convex functions of C^1 class. Fundamental theorems. Particular cases. Examples. |
| References |
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1. GOLUZIN, G. M. : Geometric theory of functions of a complex variable, Trans. Math. Mon., Amer. Math. Soc., 1969.
2. GOODMAN, A. W. : Univalent functions (vol. I, II), Mariner Publishing Co., Tampa, 1983. 3. DUREN, P. L. : Univalent functions, Springer Verlag, Berlin, Heidelberg, 1984. 4. MOCANU, PETRU - BULBOACĂ, TEODOR - SĂLĂGEAN, GR. ŞTEFAN : Teoria geometrică a funcţiilor univalente, Casa Cărţii de Ştiinţă, Cluj-Napoca, 1999. 5. BULBOACĂ, TEODOR - MOCANU, PETRU : Bevezetés az analitikus függvények geometriai elméletébe, Editura Abel (Erdely Tankönyvtanács), Cluj-Napoca, 2003. 6. GRAHAM, IAN - KOHR, GABRIELA : Geometric function theory in one and higher dimensions, M. Dekker, 2003. |
| Assessment |
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Exam. Student tests during the semester; their average represents 1/3 from the final score. |