Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 399-416

LINEAR DIFFERENTIAL EQUATIONS WITH SOLUTIONS IN THE GROWTH SPACE H_ω^∞

Juha-Matti Huusko, Taneli Korhonen and Atte Reijonen

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; juha-matti.huusko 'at' uef.fi

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; taneli.korhonen 'at' uef.fi

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; atte.reijonen 'at' uef.fi

Abstract. Sufficient conditions for solutions of

f⁽ⁿ⁾ + A_n-1(z)f^(n-1) + ⋅⋅⋅ + A₁(z)f' + A₀(z)f = A_n(z)

and their derivatives to be in H_ω^∞(D) are given by limiting the growth of coefficients A₀(z),...,A_n(z). Here H_ω^∞(D) consists of those analytic functions f in a domain D for which |f(z)|ω(z) is uniformly bounded. In particular, the case where D is the unit disc is considered. The theorems obtained generalize and improve certain results in the literature. Moreover, by using one of the main results, one can give a straightforward proof of a classical result regarding the situation where the coefficients are polynomials.

2010 Mathematics Subject Classification: Primary 34M10; Secondary 30H30.

Key words: Differential equation, unit disc, Bloch space, growth space.

Reference to this article: J.-M. Huusko, T. Korhonen and A. Reijonen: Linear differential equations with solutions in the growth space H_ω^∞. Ann. Acad. Sci. Fenn. Math. 41 (2016), 399-416.

Full document as PDF file

doi:10.5186/aasfm.2016.4128

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