Converse growth estimates for ODEs with slowly growing solutions
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Mathematische Zeitschrift
Converse growth estimates for ODEs with slowly growing solutions Janne Gröhn1 Received: 4 December 2019 / Accepted: 26 July 2020 © The Author(s) 2020
Abstract Let f 1 , f 2 be linearly independent solutions of f + A f = 0, where the coefficient A is an analytic function in the open unit disc D of the complex plane C. It is shown that many properties of this differential equation can be described in terms of the subharmonic auxiliary function u = − log ( f 1 / f 2 )# . For example, the case when supz∈D |A(z)|(1 − |z|2 )2 < ∞ and f 1 / f 2 is normal, is characterized by the condition supz∈D |∇u(z)|(1 − |z|2 ) < ∞. Different types of Blaschke-oscillatory equations are also described in terms of harmonic majorants of u. Even if f 1 , f 2 are bounded linearly independent solutions of f + A f = 0, it is possible that supz∈D |A(z)|(1 − |z|2 )2 = ∞ or f 1 / f 2 is non-normal. These results relate to sharpness discussion of recent results in the literature, and are succeeded by a detailed analysis of differential equations with bounded solutions. Analogues for the Nevanlinna class are also considered, by taking advantage of Nevanlinna interpolating sequences. It is shown that, instead of considering solutions with prescribed zeros, it is possible to construct a bounded solution of f + A f = 0 in such a way that it solves an interpolation problem natural to bounded analytic functions, while |A(z)|2 (1−|z|2 )3 dm(z) remains to be a Carleson measure. Keywords Blaschke product · Bounded solution · Growth of solution · Interpolation · Linear differential equation · Nevanlinna class · Normal function · Oscillation of solution Mathematics Subject Classification Primary 34C11; Secondary 34C10
1 Introduction Let Hol(D) be the collection of analytic functions in the open unit disc D of the complex plane C. For 0 ≤ α < ∞, let L ∞ = α denote the space of f : D → C for which f L ∞ α ∞ = H ∞ for short. supz∈D | f (z)|(1 − |z|2 )α < ∞, and write Hα∞ = L ∞ ∩ Hol(D) and H α 0 We are interested in the relation between the growth of the coefficient A ∈ Hol(D) and the
The author was supported in part by the Academy of Finland #286877.
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Janne Gröhn [email protected] Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
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oscillation and growth of solutions of f + A f = 0.
(1)
By [43, Theorems 3–4], the following conditions are equivalent: (i) A ∈ H2∞ ; (ii) zero-sequences of all non-trivial solutions ( f ≡ 0) of (1) are separated with respect to the hyperbolic metric. We refer to [3] for a far reaching generalization concerning the connection between the growth of the coefficient A ∈ Hol(D) and the minimal separation of zeros of non-trivial solutions of (1). It has been unclear whether (iii) all solutions of (1) belong to the Korenblum space 0 0. We proceed to state two generalizations in this respect. Theorem 15 in Sect. 4 shows that it is not necessary to take the infimum over the whole unit disc while Theorem 2 below
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