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

Code	Subject	
MMA1007	Univalent Functions and Differential Subordinations	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics	3	2+2+0	speciality	optional	

Teaching Staff in Charge	Prof. SALAGEAN Grigore Stefan, Ph.D., salagean@math.ubbcluj.ro	
Aims	The aim of this course is to realize a deep study of univalent functions, which are essential in geometric function theory.	
Content	1. Univalent functions. Moebius functions Applications. 
2. The class S. Covering and distortion theorems.Coefficient estimations. The compactness of the class S. 
3. Hyperbolic metric on the unit disc. Applications to univalent functions.  
4. Carathéodory&#8217;s class. Integral representation of the functions with positive real part. 
 Subordination.  Subordination principle. Sakaguchi&#8217;s Lemma
5. Starlike functions. The class  S*. 
6. Convex functions. The class K. 
7. Alpha-convex functions (Mocanu functions). Applications. 
8. Close-to-convex functions. Kaplan&#8217;s theorem. Applications. 
9. Spirallike functions. Examples and applications. 
10. Differential subordinations. Fundamental lemmas. The class of admissible functions.
11 Applications of differential subordinations theory. 
12. Subordination chains. Applications. 
13. The Loewner differential equation. 
14. Applications of subordination chains to the characterization od certain subclasses of the class S (S*, K şi C). Univalence criteria. 	
Bibliography	1. P. T. Mocanu, T. Bulboacă, G. S. Sălăgean : Teoria geometrică a funcţiilor univalente,       Cluj-Napoca: Casa Cărţii de Ştiinţă, Cluj-Napoca,Ed. II, 2006.
2. G. S. Sălăgean, Geometria planului complex, PromediaPlus, Cluj-Napoca, 1997
3. I. Graham, G. Kohr, Geometric function theory in one and higher dimensions, Marcel Inc., NY, 2003
4. Ch. Pommerenke : Univalent Functions, Göttingen: Vandenhoeck & Ruprecht,  1975
5. S. S. Miller, P. T. Mocanu : Differential Subordinations. Theory and Applications, New York &#8211; Basel, Marcel Dekker Inc., 2000
6. P. Hamburg, P. T. Mocanu, N. Negoescu, Analiză matematică (Funcţii complexe), Ed. Did. şi Ped., Bucureşti ,1982.
7. A. W. Goodman, Univalent Functions, Mariner Publ. Comp., Tampa, Florida, 1984
8. P. T. Mocanu, Funcţii complexe, Partea I, Lito. Universitaţii Cluj, 1972
9. P. Duren, Harmonic mappings in the plane, Cmbridge Univ. Press, 2004
10. A. F. Nikiforov, V. B. Uvarov, Elements de la theorie des foncti0ons speciales, Ed. Mir, Moscou, 1976
	
Assessment	exam (50%) + home-work (30%) + seminar (20%)