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+1+0
speciality
optional
Teaching Staff in Charge
Prof. SALAGEAN Grigore Stefan, Ph.D.,  salageanmath.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’s class. Integral representation of the functions with positive real part.
Subordination. Subordination principle. Sakaguchi’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’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.
References
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 – 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%)
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject