On Growth Of Subharmonic Functions In Space

On Growth of Subharmonic Functions in Space

Let u be a subharmonic function in m-dimensional real space. The study of the gr owth of u from the point of view of classical value distribution theory is a sub ject of great interest. The idea is to compare the Lp means of u on a sphere of radius r with the Reisz mass of u in that same sphere. In the two dimensional ca se, numerous results are known. But they do not all carry over to higher dimensi ons. Certain difficulties arise in higher dimensions which do not occur in dimen sion 2. For example, a subharmonic function in the Euclidean space of dimension more than 4 may fail to be square integrable over a sphere. In this thesis we propose to study such growth problems outlining the similariti es and differences between the two dimensional case and the higher dimensional c ases. The study will involve: (i) a survey of the main functionals used in class ical value distribution theory and the corresponding inequalities satisfied by t hem. (ii) definitions of corresponding functionals in the setting of subharmonic functions in m dimensional real space and statements of main theorems governing them. (iii) Calculation of spherical harmonics for a special class of such func tions. (iv) Effect of one kind of symmetrization of the Reisz mass on the size o f spherical harmonics for two classes of subharmonic functions. (v) Statement an d proof of main growth results; in particular, the L2-result when m = 2, 3, 4.

Distribution of Zeros of Entire Functions

Introduction to the Theory of Entire Functions of Several Variables