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We apply Rossi's half-plane version of Borel's Theorem to study the zerodistribution of linear combinations of $\mathcal{A}$-entire functions (Theorem1.2). This provides a unified way to study linear $q$-difference, differenceand differential operators (with entire coefficients) preserving subsets of$\mathcal{A}$-entire functions, and hence obtain several analogous results forthe Hermite-Poulain Theorem to linear finite ($q$-)difference operators withpolynomial coefficients. The method also produces a result on the existence ofinfinitely many non-real zeros of some differential polynomials of functions incertain sub-classes of $\mathcal{A}$-entire functions.

Linear $q$-difference, difference and differential operators preserving some $\mathcal{A}$-entire functions