The coefficient problem in the theory of linear partial differential equations

Abstract

1. The basic idea of the application of integral operators to the Weierstrass-Hadamard direction. In order to generate and investigate solutions of differential equations, operators p (defined as the integral operators of the first kind) have been introduced in [2; 6](2). p transforms analytic functions of one and two variables into solutions of linear elliptic differential equations of two and three variables, respectively. It has been shown in the abovementioned papers that p (as well as some other operators connected with p) preserves many properties of the functions to which the operator is applied. This situation permits us to use theorems in the theory of functions to obtain theorems not merely on harmonic functions in two variables, but on solutions of other linear differential equations as well(3). In the present paper the above-mentioned method is used to prove connections between the properties in the large of solutions i1 of certain linear differential equations, see (1.1) and (1.3), on one side and the structure of certain subsequences of the coefficients of the series development of V/ at the origin on the other. Let us formulate these procedures in a somewhat more concrete manner, at first for equations in two variables. Let i1 be a (real) solution of the differential equation