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Linear differential equations with coefficients in weighted Bergman and Hardy spaces. (English) Zbl 1133.34045
Nevanlinna theory has appeared to be a powerful tool in the field of complex differential equations. The focus on the topic that the relation between the coefficients and the solutions of linear differential equation has been studied in more detail. In this paper, firstly, the authors present some results in the unit disc \(D=\{z: | z| <1\}\), which are well-known.
Then, the differential equation
\[ f^{(k)}+a_{k-1}(z)f^{(k-1)}+\dots+a_{1}(z)f'+a_{0}(z)f=0\tag{1} \] is considered, where the coefficients \(a_{j}(z)(j=0,1,\dots,k-1)\) are analytic in a complex domain.
Secondly, the authors study the following two problems by using Nevanlinna theory:
(i) Suppose that the coefficients \(a_{j}(z)\) \((j=0,1,\dots,k-1)\) of (1) belong to some function space, such as Bergmann space \(A^{\frac{1}{k-j}}\) or weighted Hardy space \(H^{\frac{1}{k-j}}_{k-j}\) \((j=0,1,\dots,k-1)\), then the solutions of (1) belong to some function space, such as the Nevanlinna class N or the general function space F.
(ii) Suppose that the solutions of (1) belong to a certain analytic function space, such as Nevanlinna class N or the general function space F, then the coefficients \(a_{j}(z)\) \((j=0,1,\dots,k-1)\) of (1) belong to some function space.
The authors also describe the relation between the growth of solutions of (1) and the coefficients \(a_{j}(z)\) \((j=0,1,\dots,k-1)\) of (1), which belong to some function space.

MSC:
34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D55 \(H^p\)-classes (MSC2000)
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