Convex and subharmonic functions.

• Main Author: Tomiuk, Daniel.
• Other Authors: Kozakiewicz, W. (Supervisor)
• Format: Others
• Language: en
• Published: McGill University 1952
• Subjects: Functions (in Mathematics).
• Online Access: http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=123843

The principle content of this thesis could be divided roughly into three parts: a) to establish some of the more imprtant theorems of the convex and subharmonic functions; b) to give a solution fof the Dirichlet Problem for the circle which, as we will see, is essenstial in the development of the theory of subharmonic fuctions; c) to show the analogy existing between the theories of the convex and subharmonic functions. Historically the first elementary and systematic expostion of convex functions was written by J. Jensen in 1906. Of the many authors who, following Jensen, have developed the theory in its early stages we mention in particular F. Bernstein, G. Doetch, and L. Galvani [cv. Bernstein (1), *Bernstein and Doetsch (2), *and L. Galvanie, “Sulle funzioni convesse di una e duo vairabili definite in un aggregato qualunque” (Circele Mathematico di Palermo, Vol.41 1916, p.103-134)]. The theory of the subharmonic functions which was developed mainly by F. Riesz (in 1926) and Paul Montel (in 1928) appeared principally, as we will see, as a generalization of the convex functions of one variable to functions of two variables obeying the same fundamental properties [cf. F. Riesz (10) * and P. Montel (8)]*. Two other names must be mentioned in the connection of this theory, those of J.E. Littlewood and T. Rado who have further developed the general theory of Subharmonic functions [...]