Note on the distribution of zeros of polynomials

Let $\mathfrak M$ be the set of zeros of the polynomial $P(z)=\sum_{k=0}^mA_kS_k(z)$, where $S_k(z)$ are functions defined in some region $B$ and the coefficients $A_k$ are arbitrary numbers from the ring $0\leqslant r_k\leqslant|A_k-a_k|\leqslant R_k<\infty$. Conditions necessary and sufficient to ensure that $z\in\mathfrak M$ are obtained.