On complex differential equations in the unit disc.

*(English)*Zbl 0965.34075
Annales Academiæ Scientiarum Fennicæ. Mathematica. Dissertationes. 122. Helsinki: Suomalainen Tiedeakatemia. Joensuu: Univ. Joensuu, Department of Mathematics, 54 p. (2000).

In this thesis, the author considers mainly linear ordinary differential equations
\[
f^{(k)}+ A_{k-1} f^{(k- 1)}+\cdots+ A_0f= F,\tag{\(*\)}
\]
where the coefficients \(A_j\) and \(F\) are analytic functions in the unit disc. His aim is to prove estimates for the growth of the solutions to \((*)\) depending on the growth of the coefficients \(A_j\) and of \(F\) only. More precisely he looks for conditions on the \(A_j\)’s such that the solutions belong to some function spaces, e.g. to the Nevanlinna class \(N\), to the weighted Hardy space \(H^p_q\), to the \(\alpha\)-Bloch space \(B^\alpha\), or to a general function space \(F(p,q,s)\) (for definition see the paper).

In Section 1 he recalls the Nevanlinna theory in the unit disc as well as some classical results of the theory of complex differential equations. Here, he also introduces the function spaces and function classes needed later. In Section 2 he treats the first-order case, in Section 3 the second-order case of \((*)\) with \(F\equiv 0\). In Section 4 he considers the differential equation \(f^{(k)}+ Af= 0\). In Section 5 he proves upper bounds for the moduli \[ \Biggl|{f^{(n)}\over f^{(j)}}\Biggr|,\quad n> j> 0, \] for admissible meromorphic functions \(f\) in the unit disc. \(f\) is called admissible if \[ \limsup_{r\to 1-} {T(r,f)\over \log{1\over 1-r}}= \infty. \] These results are certainly of some independent interest. The general differential equation \((*)\) is considered in Section 6 and 7 where he for instance proves some results in the unit disc corresponding to results of Wittich and Frei who investigated differential equations in the complex plane. In Section 8 the author proves some results on algebraic-differential equations of first-order.

In Section 1 he recalls the Nevanlinna theory in the unit disc as well as some classical results of the theory of complex differential equations. Here, he also introduces the function spaces and function classes needed later. In Section 2 he treats the first-order case, in Section 3 the second-order case of \((*)\) with \(F\equiv 0\). In Section 4 he considers the differential equation \(f^{(k)}+ Af= 0\). In Section 5 he proves upper bounds for the moduli \[ \Biggl|{f^{(n)}\over f^{(j)}}\Biggr|,\quad n> j> 0, \] for admissible meromorphic functions \(f\) in the unit disc. \(f\) is called admissible if \[ \limsup_{r\to 1-} {T(r,f)\over \log{1\over 1-r}}= \infty. \] These results are certainly of some independent interest. The general differential equation \((*)\) is considered in Section 6 and 7 where he for instance proves some results in the unit disc corresponding to results of Wittich and Frei who investigated differential equations in the complex plane. In Section 8 the author proves some results on algebraic-differential equations of first-order.

Reviewer: Günter Frank (Berlin)

##### MSC:

34M10 | Oscillation, growth of solutions to ordinary differential equations in the complex domain |

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

30D45 | Normal functions of one complex variable, normal families |