Consider the linear differential equation
where are entire functions satisfying The following question is natural: If has no finite deficient values, does every nonconstant solution to (1) have infinite order? The authors study the growth of solutions to (1) under a condition related to this question. The main result is stated as follows: If and
as outside a set of finite logarithmic measure, then every nonconstant solution to (1) has infinite order. By T. Murai [Ann. Inst. Fourier 33, No. 3, 39-58 (1983; Zbl 0519.30029)], if has Fejér gaps, then (2) is valid for some exceptional set of finite logarithmic measure. Hence the same conclusion holds, if and has Fejér gaps.