MMA1015 | Univalent Functions |
Teaching Staff in Charge |
Prof. BULBOACA Teodor, Ph.D., bulboacamath.ubbcluj.ro |
Aims |
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 |
1. Univalent functions; classical results. (6 hours lectures + 6 hours seminar)
- 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. (6 hours lectures + 4 hours seminar + 2 hours test) - 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. (12 hours lectures + 12 hours seminar) - 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. (4 hours lectures + 2 hours seminar + 2 hours test) - 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. |
References |
- Bibliografia obligatorie:
1. Bulboacă, Teodor - Mocanu, Petru : Bevezetés az analitikus függvények geometriai elméletébe, Editura Abel, Cluj-Napoca, 2003. 2. Mocanu, Petru - Bulboacă, Teodor - Sălăgean, Gr. Ştefan : Teoria geometrică a funcţiilor univalente, ediţia a II-a, Editura Casa Cărţii de Ştiinţă, Cluj-Napoca, 2006. - Bibliografia opţională: Următoarele cărţi pot fi găsite la Biblioteca Facultăţii de Matematică şi Informatică: 1. Bulboacă, Teodor : Differential subordinations and superordinations. New Results, Editura Casa Cărţii de Ştiinţă, Cluj-Napoca, 2005. 2. Goluzin, G. M. : Geometric theory of functions of a complex variable, Trans. Math. Mon., Amer. Math. Soc., 1969. 3. Goodman, A. W. : Univalent functions (vol. I, II), Mariner Publishing Co., Tampa, 1983. 4. Duren, P. L. : Univalent functions, Springer Verlag, Berlin, Heidelberg, 1984. 5. Mocanu, Petru - Miller, S. Sanford : Differential Subordinations. Theory and Applications, M. Dekker, 2000. |
Assessment |
Exam. Student tests during the semester; their average represents 1/3 from the final score. |
Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |