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

Code	Subject	
MMC0002	Geometric Function Theory	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics - in Hungarian	4	2+1+0	speciality	optional	
Mathematics - in Romanian	6	2+1+0	speciality	optional	
Mathematics-Computer Science - in Hungarian	4	2+1+0	speciality	optional	
Applied Mathematics	6	2+1+0	speciality	optional	

Teaching Staff in Charge	Prof. SALAGEAN Grigore Stefan, Ph.D., salagean@math.ubbcluj.ro
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. 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.	
Bibliography	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	Exam. Student tests during the semester; their average represents 1/3 from the final score.