Fast Growing Solutions of Linear Differential Equations in the Unit Disc

Abstract

For $n \in \mathbb{N}$, the n-order of an analytic function f in the unit disc D is defined by

$$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$

where log⁺ x  =  max{log x, 0}, log⁺₁ x  =  log⁺ x, log⁺_n+1 x  =  log⁺ log⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation

$$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$

where the coefficients are analytic in D, satisfy σ_M,n+1(f)  ≤  α if and only if σ_M,n(a _j )  ≤  α for all j  =  0, ..., k − 1. Moreover, if q ∈{0, ..., k − 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$, then there are at least k − q linearly independent solutions f of ($\dag$) such that σ_M,n+1(f) = σ_M,n(a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.