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

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 \begin{equation} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots +A_1(z)f’+A_0(z)f=0,\tag {\dagger } \end{equation} 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 \begin{equation*} \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 \}, \end{equation*} where $p_k:=0$ and \begin{equation*} \sigma _M(f):=\limsup _{r\to 1^-}\frac {\log ^+\log ^+ M(r,f)}{-\log (1-r)}. \end{equation*} In addition, if $n\in \{0,\ldots ,k-1\}$ is the smallest index for which \begin{equation*} \frac {p_n}{k-n} = \max _{j=0,\ldots ,k-1} \frac {p_j}{k-j}, \end{equation*} then there are at least $k-n$ linearly independent solutions of $(\dagger )$ such that \begin{equation*} \sigma _M(f)\geq \max _{j=0,\ldots ,k-1} \frac {p_j}{k-j} - 2. \end{equation*} These results are a generalization of a recent result due to Chyzhykov, Gundersen and Heittokangas.