The authors study the growth of solutions of the linear periodic differential equation
and its corresponding homogeneous equation
where are polynomials in . It is well-known that all solution of (2) are entire functions. Let denote the order of growth of and use the notation to denote the hyper-order of .
Theorem 1. If
then the differential equation (2) has no nontrivial subnormal solution, and every solution of (2) satisfies .
Theorem 2. If and () satisfy (3), then (i) (1) has at most one subnormal solution , and , where and are polynomials in ; (ii) all other solutions of (1) satisfy except a subnormal solution in (i).
Theorem 2 refines and generalizes a result due to G. Gundersen and M. Steinbart for .