Abstract: Nevanlinna theory has been widely applied in the field of complex differential equa-tions. It is an important research subject to explore the relationship between the growth of the coefficients and the growth and value distribution of meromorphic solutions of complex linear differential equations by Nevanlinna theory. Meanwhile, the gap series has some special properties which may play important roles when the gap series appear as the coefficients of certain equation. Therefore, the properties of meromorphic solutions of complex linear differential equations can be investigated by combining with the definition and properties of gap series. In this paper, we consider a kind of the homogeneous and non-homogeneous higher order complex linear differential equation based on Nevanlinna theory and the definition and properties of Fejér gap series. When one of the coefficients is relative to Fejér gap series and the others are entire or meromorphic functions, the estimates on the order of meromorphic solutions of the involved equation are obtained, which promotes and improves the previous research results.

Key words: complex linear differential equation, Nevanlinna theory, Fejér gap series, iterated order, iterated type