Under review the author proves the following ``Picard type'' theorem and a corresponding normality criterion: (1) Let \(f\) be an entire function, \(n\geq 2\), \(k\geq 1\) and \(a,a_1,\dots, a_m\in\mathbb{C}\) with \(a\neq 0\) and let \[ P[u]:= \sum^m_{j=1}a_j \prod^{s_j}_{\nu=1} u^{(k_\nu^{(j)})} \] be a differential polynomial with \(2\leq s_j\leq n-1\), \(\sum^{s_j}_{\nu=1} (k_\nu^{(j)}) \geq 1\) for \(j=1,2,\dots,m\). If \(\psi:=f^n+ af^{(1)}+ P[f]\) has no zeros in \(\mathbb{C}\), then \(f\) is constant. (2) Let \({\mathcal F}\) be a family of functions analytic in \(\mathbb{D}\), \(n\geq 2\), \(k\geq 1\) and let \(a,b,a_1,\dots,a_m\) be meromorphic in \(\mathbb{D}\) with \(a\neq 0\), where all poles of \(a\) have multiplicity at most \(n-1\). Let \(P[u]\) be the differential polynomial of (1) with \[ (n-1) \sum^{s_j}_{\nu=1} k_\nu^{(j)}+ k s_j\leq kn \] for all \(j=1,2,\dots,m\) where equality can hold only if \(2\leq s_j\leq n-1\). If for all \(f\in{\mathcal F}\) and for all \(z\in\mathbb{D}\) we have \[ a(z)f^n(z)+ f^{(k)}(z)+ P[f](z) -b(z)\neq 0, \] then \({\mathcal F}\) is normal. These results generalize results of Hayman, Drasin, Langley, Chen and Hua.