Fast growing solutions of linear differential equations in the unit disc.

*(English)*Zbl 1120.34071In several recent papers, the authors have studied solutions of complex differential equations in the unit disc \(\mathbb{D}\), see e.g. [Ann. Acad. Sci. Fenn. Math. Diss. 122, 1–54 (2000; Zbl 0965.34075), Ann. Acad. Sci. Fenn. Math. 29, 233–246 (2004; Zbl 1057.34111), Proc. Am. Math. Soc. 135, No. 5, 1355–1363 (2007; Zbl 1122.34067)], making use of notions from complex function spaces to control solutions that have their growth comparable to that of the coefficients. This paper is devoted to considering fast growing solutions, in which case one may expect to obtain results similar to those valid in the whole plane. The authors here make use of iterated logarithms to define the \(n\)-order of an analytic function \(f\) as

\[ \sigma_{M,n}(f):=\limsup_{r\to1^-}\frac{\log^+_{n-1}M(r,f)}{-\log(1-r)}; \]

here, \(\log^+_1x:=\max(\log x,0)\) and \(\log_{n+1}^+x=\log^+\log^+_nx\).

Several results are now proved, together with a number of examples showing the sharpness of the results obtained. As an example of the results, consider solutions \(f\) of \[ f^{(k)}+a_{k-1}(z)f^{(k-1)}+\dots+a_1(z)f'+a_0(z)f=0\tag{*} \] with coefficients analytic in \(\mathbb{D}\). Then \(\sigma_{M,n+1}(f)\leq\alpha\) if and only if \(\sigma_{M,n}(a_j)\leq\alpha\) holds for all \(j=0,\dots,k-1\). Moreover, if \(q\) is the largest index for which \(\sigma_{M,n}(a_q)=\max_{0\leq j\leq k-1}\sigma_{M,n}(a_j)\), then there are at least \(k-q\) linearly independent solutions \(f\) of (*) such that \(\sigma_{M,n+1}(f)=\sigma_{M,n}(a_q)\). The proofs in this paper strongly rely on logarithmic derivative estimates in the unit disc, see the estimates by the first author, G. G. Gundersen and I. Chyzhykov [Proc. London Math. Soc. 86, No. 3, 735–754 (2003; Zbl 1044.34049)] and their refinements in this paper.

We finally remark that a closely related, independent, paper by T. Cao and H. Yi [J. Math. Anal. Appl. 319, 278–294 (2006; Zbl 1105.34059)] contains similar, and partially same, results as in the present paper.

\[ \sigma_{M,n}(f):=\limsup_{r\to1^-}\frac{\log^+_{n-1}M(r,f)}{-\log(1-r)}; \]

here, \(\log^+_1x:=\max(\log x,0)\) and \(\log_{n+1}^+x=\log^+\log^+_nx\).

Several results are now proved, together with a number of examples showing the sharpness of the results obtained. As an example of the results, consider solutions \(f\) of \[ f^{(k)}+a_{k-1}(z)f^{(k-1)}+\dots+a_1(z)f'+a_0(z)f=0\tag{*} \] with coefficients analytic in \(\mathbb{D}\). Then \(\sigma_{M,n+1}(f)\leq\alpha\) if and only if \(\sigma_{M,n}(a_j)\leq\alpha\) holds for all \(j=0,\dots,k-1\). Moreover, if \(q\) is the largest index for which \(\sigma_{M,n}(a_q)=\max_{0\leq j\leq k-1}\sigma_{M,n}(a_j)\), then there are at least \(k-q\) linearly independent solutions \(f\) of (*) such that \(\sigma_{M,n+1}(f)=\sigma_{M,n}(a_q)\). The proofs in this paper strongly rely on logarithmic derivative estimates in the unit disc, see the estimates by the first author, G. G. Gundersen and I. Chyzhykov [Proc. London Math. Soc. 86, No. 3, 735–754 (2003; Zbl 1044.34049)] and their refinements in this paper.

We finally remark that a closely related, independent, paper by T. Cao and H. Yi [J. Math. Anal. Appl. 319, 278–294 (2006; Zbl 1105.34059)] contains similar, and partially same, results as in the present paper.

Reviewer: Ilpo Laine (Joensuu)