Growth of solutions of an n-th order linear differential equation with entire coefficients

Abstract

We consider a differential equation f⁽ⁿ⁾+A_n−1(z)f⁽ⁿ⁻¹⁾+…+A₁(z)f'+A₀(z)f=0, where A₀(z), ..., A_n−1(z) are entire functions with A₀(z){¬≡}0. Suppose that there exist a positive number μ, and a sequence (z_j)_j∈N with lim_j→+∞z_j=∞, and also two real numbers α, β (0≤β<α) such that |A₀(z_j)|≥e^α|z_j|μ and |A_k(z_j)|≤e^β|z_j|μ as j→+∞ (k=1, ..., n−1). We prove that all solutions f{¬≡}0 of this equation are of infinite order. This result is a generalization of one theorem of Gundersen ([3], p. 418).