 Let $\mathcal F$ be a family of holomorphic functions in a complex domain $D$ such that for every $f\in\mathcal F$ and for every $m\in\{0,\dots,q\},$ all zeros of $f^n(f^{n_0})^{(t_0)}\cdots(f^{n_k})^{(t_k)}-a_m$ have multiplicity at least $\ell_m.$ Assume that  $ a)\quad n_j\geq t_j \text{\;for all \;} 0\leqslant j\leqslant k,\;\text{ and\;} \ell_i\geq 0 \text{\;for all\;} 0\leqslant i\leqslant q,$\ $ b)\quad \sum_{i=0}^q\frac{0}{\ell_i}<\frac{ qn-0+\sum_{j=0}^kq(n_j-t_j)}{n+\sum_{j=0}^k