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Linear differential equations in the unit disc with analytic solutions of finite order. (English) Zbl 1122.34067

A function \(g\), analytic in the unit disc \(D\), belongs to the weighted Hardy space \(H_q^{\infty}\) if \[ \sup_{0\leq r<1}M(r,g)(1-r^2)^q <\infty, \]
where \(M(r,g)\) is the maximum modulus of \(g(z)\) in the circle of radius \(r\) centered at the origin. If \(g\) belongs to \(H_q^{\infty}\) for some \(q\geq 0,\) then it is said to be an \(\mathcal{H}-\)function. Heittokangas has shown that all solutions of the linear differential equation \[ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f'+ A_0(z)f=0,\tag{1} \]
where \(A_j(z)\) is analytic in \(D\) for all \(j=0,\dots,k-1,\) are of finite order of growth in \(D\) if and only if all coefficients \(A_j(z)\) are \(\mathcal{H}-\)functions.
It is said that \(g\in G_p\) when \(p=\inf\{q\geq 0: g\in H_q^{\infty}\}.\) The authors prove that if all coefficients \(A_j(z)\) of (1) satisfy \(A_j\in G_{p_j}\) for all \(j=0,\dots,k-1,\) then all nontrivial solutions \(f\) of (1) satisfy \[ \min_{j=1,\dots,k}\frac{p_0-p_j}j-2\leq \sigma_M(f)\leq\max_{j=0,\dots,k-1}\{0,\frac{p_j}{k-j}-1\}, \] where \(p_k=0\) and \[ \sigma_M(f)=\limsup_{r\to 1^-}\frac{\log^+\log^+M(r,f)}{-\log(1-r)}. \] In addition, if \(n\in\{0,\dots,k-1\}\) is the smallest index for which \[ \frac{p_n}{k-n}=\max_{j=0,\dots, k-1}\frac{p_j}{k-j}, \] then there are at least \(k-n\) linearly independent solutions of (1) such that \[ \sigma_M(f)\geq \max_{j=0,\dots,k-1}\max_{j=0,\dots,k-1}\frac{p_j} {k-j}-2. \] These results are generalizations of a result due to Chyzhykov-Gundersen-Heittokangas.

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
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References:

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