다음은 T. Rado 의 책 On the Problem of Plateau / Subharmonic Functions 의 서문이다.
A convex function $l$ may be called sublinear in the following sense: if a linear function $l$ is bigger than or equal to $f$ at the boundary points of an interval, the $l\geq f$ in the interior of that interval also. If we replace the terms of interval and linear function by the terms domain and harmonic function, we obtain a statement which expresses the characteristic property of subharmonic functions of two or more variables. This generalization, formulated and developed by F. Riesz, immediately attracted the attention of many mathematicians, both on account of its intrinsic interest and on account of the wide range of its applications. If $f(z)$ is analytic function of the complex variable $z=x+iy$, then $|f(z)|$ is subharmonic. The potential of a negative mass-distribution is subharmonic. in differential geometry, surfaces of negative curvature and minimal surfaces can be characterized in terms of subharmonic functions.The idea of a subharmonic function leads to significant applications and interpretations in the fields just referred to, and conversely, every one of these fields is and apparently inexhaustible source of new theorems on subharmonic functions, either by analogy or by direct implication. The purpose of this report is first to give a detailed account of those facts which seem to constitute the general theory of subharmonic functions, and second to present a selected groups of facts which seem to be well adapted to illustrate the relationships between subharmonic functions and other theories. Roughly, Chapters $I,II,III,V,VI$ are devoted to the first purpose, while chapters $IV,VII$ are devoted to the second one-The presentation is formulated for the case of two independent variables,but both the methods and the results remain valid in the general case, except for obvious modifications, unless the contrary is explicitly stated.
Subharmonic functions have a long and interesting history. F. Riesz points that various methods duo to Poincare, Perron, Remak in portential theory and to Hartogs and R. Nevanlinna in the theory of functions of functions a complex variable, are based essentially on the idea of a subharmonic function. The reader should consult Riesz's books for detailed historical references. Readers interested in the possibilities of applying subharmonic functions mmay read, for general information, Riesz, Evans, Frostman.
As it has been observed above, potentials of negative mass-distributions are subharmonuc functions, and essentially the converse is also true. Thus the theory of subharmonuc functions may be interpreted as the study of such potentials based on a few characteristic properties, while the methods of potential theory are based on the representation in terms of definite integrals. It is very probable that the range of the theory of subharmonic functions, interpreted in this manner, will be considerably extended in the near future. For instance, the sweeping-out process, which is fundamental in the recent development of the theory of the capacity of sets, could be easily interpreted in terms of harmonic majorants of subharmonic functions.
Historically, the first generalization of convex functions of a single variable is represented by the convex functions of several variables, characterized by the property of being sublinear on every straight segment within the domain of definition. While such functions are easily seen to be subharmonic, their theory was developed in connection with problems of an entirely different type. For this reason, the theory of these functions will be included among the topics discussed by W. Fenchel in a subsequent report of this series.
The Ohio State University, March 1937.
Tibor Rado.