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

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

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

()

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

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

where $p_k:=0$ and

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

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

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

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

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

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