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

Abstract

A function g g , analytic in the unit disc D D , belongs to the weighted Hardy space H q ∞ H_q^\infty if sup 0 ≤ r > 1 M ( r , g ) ( 1 − r 2 ) q > ∞ \sup _{0\le r>1}M(r,g)(1-r^2)^q>\infty , where M ( r , g ) M(r,g) is the maximum modulus of g ( z ) g(z) in the circle of radius r r centered at the origin. If g g belongs to H q ∞ H_q^\infty for some q ≥ 0 q\geq 0 , then it is said to be an H \mathcal {H} -function. Heittokangas has shown that all solutions of the linear differential equation (\dagger) f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 , \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 ) A_j(z) is analytic in D D for all j = 0 , … , k − 1 j=0,\ldots ,k-1 , are of finite order of growth in D D if and only if all coefficients A j ( z ) A_j(z) are H \mathcal {H} -functions. It is said that g ∈ G p g\in G_p when p = inf { q ≥ 0 : g ∈ H q ∞ } p=\inf \{q\geq 0 : g\in H^\infty _q\} . In this study it is shown that if all coefficients A j ( z ) A_j(z) of ( † ) (\dagger ) satisfy A j ∈ G p j A_j\in G_{p_j} for all j = 0 , … , k − 1 j=0,\ldots ,k-1 , then all nontrivial solutions f f of ( † ) (\dagger ) satisfy min j = 1 , … , k p 0 − p j j − 2 ≤ σ M ( f ) ≤ max { 0 , max j = 0 , … , k − 1 p j k − j − 1 } , \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 p_k:=0 and σ M ( f ) := lim sup r → 1 − log + ⁡ log + ⁡ M ( r , f ) − log ⁡ ( 1 − r ) . \begin{equation*} \sigma _M(f):=\limsup _{r\to 1^-}\frac {\log ^+\log ^+ M(r,f)}{-\log (1-r)}. \end{equation*} In addition, if n ∈ { 0 , … , k − 1 } n\in \{0,\ldots ,k-1\} is the smallest index for which p n k − n = max j = 0 , … , k − 1 p j k − j , \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 k-n linearly independent solutions of ( † ) (\dagger ) such that σ M ( f ) ≥ max j = 0 , … , k − 1 p j k − j − 2. \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.