Zero distribution of some shift polynomials

Given an entire function f of finite order ρ, let g(f):=∑j=1kbj(z)f(z+cj) be a shift polynomial of f with small meromorphic coefficients bj in the sense of O(rλ+ε)+S(r,f), λ<ρ. Provided α, β, b0 are similar small meromorphic functions, we consider zero distribution of fn(g(f))s−b0, resp. of g(f)−αfn−β.