The notions of (α, β, γ)- order and (α, β, γ)- type have been introduced by B. Belaïdi and T. Biswas, as tools for the investigation of the growth of the solutions of linear differential equations with meromorphic coefficients. In this paper, the notions of (α, β, γ)- order and (α, β, γ)- type are used to estimate the growth of the Nevanlinna characteristic of a differential polynomial generated by a meromorphic function by comparison with the Nevanlinna characteristic of a composition of this function with another meromorphic function, one of these functions being entire function.