Zero distribution of some shift polynomials

Given an entire function f of finite order ρ, let $g(f):={\sum }_{j=1}^k{b}_j(z)f(z+c_j)$ be a shift polynomial of f with small meromorphic coefficients $b_j$ in the sense of $O(r^{\lambda +\epsilon })+S(r,f)$, $\lambda <\rho$. Provided α, β, $b_0$ are similar small meromorphic functions, we consider zero distribution of $f^n{(g(f))}^s?b_0$, resp. of $g(f)?\alpha f^n?\beta$.