  We apply Rossi's half-plane version of Borel's Theorem to study the zero
distribution of linear combinations of $\mathcal{A}$-entire functions (Theorem
1.2). This provides a unified way to study linear $q$-difference, difference
and differential operators (with entire coefficients) preserving subsets of
$\mathcal{A}$-entire functions, and hence obtain several analogous results for
the Hermite-Poulain Theorem to linear finite ($q$-)difference operators with
polynomial coefficients. The method also produces a result on the existence of
infinitely many non-real zeros of some differential polynomials of functions in
certain sub-classes of $\mathcal{A}$-entire functions.
