Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 617-629

SLOWLY GROWING SOLUTIONS OF ODES REVISITED

Janne Gröhn

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, FI-80101 Joensuu, Finland; janne.grohn 'at' uef.fi

Abstract. Solutions of the differential equation f'' + Af = 0 are considered assuming that A is analytic in the unit disc D and satisfies

(*) sup_z∈D|A(z)|(1 – |z|²)² log e/(1 – |z|) < ∞.

By recent results in the literature, such restriction has been associated to coefficient conditions which place all solutions in the Bloch space B. In this paper it is shown that any coefficient condition implying (*) 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 B? An overall revised look into slowly growing solutions is presented, emphasizing function spaces B, BMOA, VMOA.

2010 Mathematics Subject Classification: Primary 34C10; Secondary 30D45.

Key words: Bloch space, BMOA, growth of solution, linear differential equation, oscillation of solution, VMOA.

Reference to this article: J. Gröhn: Slowly growing solutions of ODEs revisited. Ann. Acad. Sci. Fenn. Math. 43 (2018), 617-629.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4339