Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Analiză reală şi complexă (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MT255	Univalent Functions and Differential Subordinations	

Semester	Hours: C+S+L	Category	Status	
1	2+1+1		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
SALAGEAN Grigore Stefan	Ph.D.	Prof.	Function Theory	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
The aim of this course is to realize a deep study of univalent functions, which are essential in geometric function theory.	

IV. Course contents
1. Univalent functions. Area Theorem. Covering and distortion theorems.
2. Holomorphic functions with positive real part. Herglotz formula. Integral representations. Subordination.
3. Classes of univalent functions. Starlike functions, convex functions, alpha-convex functions, spirallike functions, typically real functions. Meromorphic functions.
4. Differential subordinations. Fundamental lemmas. The class of admissible functions. Applications.

V. Bibliography
1. I. Graham, G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker Inc., New York, 2003.
2. S. S. Miller, P. T. Mocanu, Differential Subordinations. Theory and Applications, New York – Basel, Marcel Dekker Inc., 2000.
3. Ch. Pommerenke, Univalent Functions, Göttingen: Vandenhoeck & Ruprecht,  1975.
4. A. W. Goodman, Univalent Functions, Mariner Publ. Comp., Tampa, Florida, 1984.
5. P. T. Mocanu, T. Bulboacă, G. S. Sălăgean, Teoria Geometrică a Funcţiilor Univalente, Cluj-Napoca: Casa Cărţii de Ştiinţă, Cluj-Napoca, Ed. II, 2006.
6. G. Kohr, P.T. Mocanu, Capitole Speciale de Analiză Complexă, Presa Univ. Clujeană, Cluj-Napoca, 2005.
7. P. Duren, Univalent Functions, Springer-Verlag, New York, 1983
8. W. Rudin, Real and Complex Analysis, 3nd ed., Mc. Graw-Hill, 1987.
9. G. S. Sălăgean, Geometria Planului Complex, Promedia Plus, Cluj-Napoca, 1997.
10. P. Hamburg, P. T. Mocanu, N. Negoescu, Analiză Matematică (Funcţii complexe), Ed. Did. şi Ped., Bucureşti ,1982.	

VI. Thematic of didactic activities per weeks
Cursuri
1. Univalent functions. Applications. [5, Cap. 1], [9, Cap. 3], [10, Cap. 6], [1, Cap. 1].
2. The class S. Covering and distortion theorems.Coefficient estimations. The compactness of the class S. [5, Cap. 1], [9, Cap. 3], [1, Cap. 1]. 
3. Hyperbolic metric on the unit disc.. Applications to univalent functions.  [9, Cap. 2],  [1, Cap. 1]. 
4. Carathéodory’s class. Integral representation of the functions with positive real part. 
 Subordination. [5, Cap. 3], [1, Cap. 2], [7, Cap. 1]. 
5. Starlike functions. The class  S*. [5, Cap. 4], [1, Cap. 2], [7, Cap. 2]. 
6. Convex functions. The class K. [5, Cap. 4], [1, Cap. 2], [7, Cap. 2]. 
7. Alpha-convex functions (Mocanu functions). Applications. [5, Cap. 4], [1, Cap. 2].
8. Close-to-convex functions. Kaplan’s theorem. Applications. [5, Cap. 4], [1, Cap. 2], [7, Cap. 2]. 
9. Spirallike functions. Examples and applications. [5, Cap. 4], [1, Cap. 2], [7, Cap. 2]. 
10. Differential subordinations. Fundamental lemmas. The class of admissible functions. [5, Cap. 10], [2, Cap. 2]. 
11 Applications of differential subordinations theory. [5, Cap. 10], [2, Cap. 2-4]. 
12. Subordination chains. Applications. [5, Cap. 8], [1, Cap. 3], [3, Cap. 6]. 
13. The Loewner differential equation. [1, Cap. 3], [3, Cap. 6].    
14. Applications of subordination chains to the characterization od certain subclasses of the class 
S (S*, K şi C). Univalence criteria. [1, Cap. 3], [3, Cap. 6]. 
 
Seminarii
1.  Moebius functions; applications. [9, Cap.2], [10, Cap. 2]
2.  Conformal mappings; applications. [10, Cap. 6], [9, Cap. 2], [6, Cap. 5],
3.  Univalent functions; applications, examples [1, Cap. 1], [6, Cap. 1],  [2, Cap. 4],
4. Subordination principle. Sakaguchi’s Lemma. Applications to the clsses S* and K. [5, Cap. 3-4], [1, 
Cap. 2], [3, Cap. 3]. 
5. Applications to the clsses of spirallike, close-to-convex and alpha-convex functions ; problems. [5, Cap. 3-4], [1, Cap. 2]. 
6. Applications  of the differential subordinations theory - examples, problems [5, Cap. 10], [2, Cap. 2-4]. 
7.  Applications of the subordination chains - examples, problems, univalence criteria. [1, Cap. 3], [9, Cap. 6].	

VII. Didactic methods used

VIII. Assessment
exam (70%) + home-work (20%) + seminar (10%)	

IX. Additional references
1. C.A. Berenstein, C.A., Gay, R., Complex Variables: An Introduction, Springer-Verlag New York, 1991. 
2. M. Rosenblum, J. Rovnyak, Topics in Hardy Classes and Univalent Functions, Birkhäuser Verlag, 1994
3. T. Ceauşu, N. Suciu, Funcţii Complexe. Probleme şi Exerciţii, Editura Mirton, Timişoara, 2001.
4. Henrici, P. Applied and Computational Complex Analysis, I-III, Wiley Classical Library, J. Wiley & Sons, New York, 1993.
5. D. J. Hallenbeck, T. H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman Adv. Publ. Program, Boston-London-Melbourne, 1984
6.  R. Remmert, Theory of Complex Functions, Springer-Verlag, New York, 1991. 
7. T. Bulboacã, Differential Subordinatios and Superordinations. Recent Results. Casa Cărţii de Ştiinţă, Cluj-Napoca, 2005. 
8. R. Narasimhan, Y. Nievergelt, Complex Analysis in One Variable, Second Edition, Birkhäuser, 1985.
9. O. Mayer, Teoria Funcţiilor de o Variabilã Complexã, Editura Academiei Române, Bucureşti, 1981. 
10. J.B. Conway, Functions of One Complex Variable, vol. I-II, Graduate Texts in Mathematics, 159, 
Springer Verlag, New York, 1996.	

	Dean,							Chair head,
	Prof. Leon TAMBULEA, Ph.D            		Prof. Grigore Stefan SALAGEAN, Ph.D
	
	Date,	09.03.2009					Course responsible,
							            Prof. Grigore Stefan SALAGEAN, Ph.D.