Infinite Order of The Solutions of Higher Order Homogeneous Linear Differential Equations with Entire Coefficients Having Completely Regular Growth Property and Bounded Components Fatou Set

The homogeneous higher order complex linear differential equations (n-thCLDEs) with entire functions is considered in this paper. We investigated some conditions that can be put on the coefficients which guarantee that any nonzero solution of such equations has infinite order. The conditions we stated are the completely regular growth (CRG), the characteristic function of some coefficients is approximately equals to the logarithm of its maximum modulus and the Denjoy’s conjecture (DC) property.