Normal Families Concerning Exceptional Functions of Derivatives

Let l be a positive integer, \(A > 1\) $A>1$ be a constant, and \(\varphi (z)\) $\phi (z)$ ( \(\not \equiv 0\) $\not\equiv 0$ ) be a function holomorphic in a domain D, all of whose zeros have multiplicity at most l, and let \({\mathcal {F}}\) $\mathcal{F}$ be a family of meromorphic functions defined in D. If, for every function \(f\in {\mathcal {F}}\) $f\in \mathcal{F}$ , (a) all zeros of f are multiple, and \(f(z)=0 \Rightarrow |f''(z)| \le A|\varphi (z)|\) $f(z)=0\Rightarrow |f^{\prime \prime }(z)|\le A|\phi (z)|$ ; (b) \(f''(z) \ne \varphi (z)\) $f^{\prime \prime }(z)\ne \phi (z)$ ; (c) all poles of f have multiplicity at least \(l+3\) $l+3$ , then \({\mathcal {F}}\) $\mathcal{F}$ is normal in D. Also, we give an example to show that the number \(l+3\) $l+3$ is sharp, and prove that the counterexample is unique in some sense.