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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Linear differential equations in the unit disc with analytic solutions of finite order


Authors: Risto Korhonen and Jouni Rättyä
Journal: Proc. Amer. Math. Soc. 135 (2007), 1355-1363
MSC (2000): Primary 34M10; Secondary 30D35
Published electronically: October 27, 2006
MathSciNet review: 2276644
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Abstract: A function $ g$, analytic in the unit disc $ D$, belongs to the weighted Hardy space $ H_q^\infty$ if $ \sup_{0\le 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

$\displaystyle f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots +A_1(z)f'+A_0(z)f=0,$ (\dag)

where $ A_j(z)$ is analytic in $ D$ for all $ j=0,\ldots,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^\infty_q\}$. In this study it is shown that if all coefficients $ A_j(z)$ of $ (\dagger)$ satisfy $ A_j\in G_{p_j}$ for all $ j=0,\ldots,k-1$, then all nontrivial solutions $ f$ of $ (\dagger)$ satisfy

$\displaystyle \min_{j=1,\ldots,k} \frac{p_0-p_j}{j}-2 \le \sigma_M(f) \le \max\left\{0, \max_{j=0,\ldots,k-1} \frac{p_j}{k-j}-1\right\}, $

where $ p_k:=0$ and

$\displaystyle \sigma_M(f):=\limsup_{r\to1^-}\frac{\log^+\log^+ M(r,f)}{-\log(1-r)}. $

In addition, if $ n\in\{0,\ldots,k-1\}$ is the smallest index for which

$\displaystyle \frac{p_n}{k-n} = \max_{j=0,\ldots,k-1} \frac{p_j}{k-j}, $

then there are at least $ k-n$ linearly independent solutions of $ (\dagger)$ such that

$\displaystyle \sigma_M(f)\geq \max_{j=0,\ldots,k-1} \frac{p_j}{k-j} - 2. $

These results are a generalization of a recent result due to Chyzhykov, Gundersen and Heittokangas.


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Additional Information

Risto Korhonen
Affiliation: Department of Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland
Email: risto.korhonen@joensuu.fi

Jouni Rättyä
Affiliation: Department of Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland
Email: jouni.rattya@joensuu.fi

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08581-9
PII: S 0002-9939(06)08581-9
Received by editor(s): June 15, 2005
Received by editor(s) in revised form: November 22, 2005
Published electronically: October 27, 2006
Additional Notes: The research reported in this paper was supported in part by the Academy of Finland grant numbers 204819 and 210245 and by the MEC Spain MTM2005-07347
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2006 American Mathematical Society