Geometric function theory 
ter 

Teaching Staff in Charge 
 
 

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. 
References 
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, ClujNapoca, 1999. 5. BULBOACĂ, TEODOR  MOCANU, PETRU : Bevezetés az analitikus függvények geometriai elméletébe, Editura Abel (Erdely Tankönyvtanács), ClujNapoca, 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. 