Zero distribution of solutions of complex linear differential equations determines growth of coefficients

It is shown that the exponent of convergence λ(f) of any solution f of $$ f^(k)+A_k-2(z)f^(k-2)+⋅s +A_1(z)f^′ +A_0(z)f=0,\quad k≥ 2, $$ with entire coefficients A0(z), …, Ak−2(z), satisfies λ(f) ⩽ λ ∈ [1, ∞) if and only if the coefficients A0(z), …, Ak−2(z) are polynomials such that $ \deg (A_j)≤ (k-j)(λ -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