Slowly growing solutions of ODEs revisited
| Autor: | Janne Gröhn |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Bloch space
Mathematics - Complex Variables Differential equation Function space General Mathematics 010102 general mathematics Ode 01 natural sciences Primary 34C10 Secondary 30D45 Combinatorics Mathematics - Classical Analysis and ODEs 0103 physical sciences Classical Analysis and ODEs (math.CA) FOS: Mathematics 010307 mathematical physics Complex Variables (math.CV) 0101 mathematics Unit (ring theory) Mathematics |
| Zdroj: | Annales Academiae Scientiarum Fennicae Mathematica. 43:617-629 |
| ISSN: | 1798-2383 1239-629X |
| DOI: | 10.5186/aasfm.2018.4339 |
| Popis: | Solutions of the differential equation $f''+Af=0$ are considered assuming that $A$ is analytic in the unit disc $\mathbb{D}$ and satisfies \begin{equation} \label{eq:dag} \sup_{z\in\mathbb{D}} \, |A(z)| (1-|z|^2)^2 \log\frac{e}{1-|z|} < \infty. \tag{$\star$} \end{equation} By recent results in the literature, such restriction has been associated to coefficient conditions which place all solutions in the Bloch space $\mathcal{B}$. In this paper it is shown that any coefficient condition implying \eqref{eq:dag} fails to detect certain cases when Bloch solutions do appear. The converse problem is also addressed: What can be said about the growth of the coefficient $A$ if all solutions of $f''+Af=0$ belong to $\mathcal{B}$? An overall revised look into slowly growing solutions is presented, emphasizing function spaces $\mathcal{B}$, $\rm{BMOA}$ and $\rm{VMOA}$. 14 pages |
| Databáze: | OpenAIRE |
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