Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMA1015	Univalent Functions	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics Computational - in Hungarian	4	2+2+0	speciality	optional	
Didactic Mathematics - in Hungarian	4	2+2+0	speciality	optional	

Teaching Staff in Charge	Prof. BULBOACA Teodor, Ph.D., bulboaca@math.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.	
Bibliography	- 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.