Containment Regions for Zeros of Analytic Functions

Abstract

In the first part of the paper Containment regions are given for the numerical ränge of some special linear Operators in which are defined from infinite bordered diagonal matrices. In the second part analytic (entire) functions are considered, which are linearized via bounded (compact) linear Operators. The application of a Gersgorin theorem, Weyl's inequality for the eigenvalues and Singular values of a compact linear operator, and the results of the first part on numerical ranges of linear Operators give Containment regions for the zeros of analytic (entire) functions. In the third part certain meromorphic functions are considered which are linearized via certain unbounded linear Operators. Again the results of the first part on numerical ranges of linear Operators give Containment regions for the zeros of these meromorphic functions.