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Zero distribution of solutions of complex linear differential equations determines growth of coefficients
Author(s) -
Heittokangas Janne,
Rättyä Jouni
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201010038
Subject(s) - mathematics , exponent , zero (linguistics) , pure mathematics , convergence (economics) , distribution (mathematics) , mathematical analysis , unit (ring theory) , differential equation , linear differential equation , combinatorics , philosophy , linguistics , mathematics education , economics , economic growth
It is shown that the exponent of convergence λ( f ) of any solution f of$$ f^{(k)}+A_{k-2}(z)f^{(k-2)}+\cdots +A_1(z)f^{\prime }+A_0(z)f=0,\quad k\ge 2, $$ with entire coefficients A 0 ( z ), …, A k −2 ( z ), satisfies λ( f ) ⩽ λ ∈ [1, ∞) if and only if the coefficients A 0 ( z ), …, A k −2 ( z ) are polynomials such that $ \deg (A_j)\le (k-j)(\lambda -1) $ for j = 0, …, k − 2. In the unit disc analogue of this result certain intersections of weighted Bergman spaces take the role of polynomials. The key idea in the proofs is W. J. Kim’s 1969 representation of coefficients in terms of ratios of linearly independent solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim