On row differential inequalities related to normality and quasi-normality (Q6618168)

The aim of the paper is to give new necessary conditions of normality and quasi-normality of families of functions in terms of differential inequalities. The main results are the following\N\NTheorem 2.3. Let \(C>0\), \(k \in \mathbb{N}\), and let \(a_0(z), \ldots, a_{k-1}(z)\) be fixed holomorphic functions in a domain \(D\). Then the family \(\mathcal{F}\) of all functions holomorphic in a domain \(D\) such that\N\[\N|f^{(k)}(z)+ a_{k-1}(z)f^{(k-1)}(z)+\ldots+a_0(z)f(z)| < C , \qquad f \in \mathcal{F}, \, z \in D,\N\]\Nis quasi-normal in \(D\).\N\NTheorem 2.4. Let \(A, B \in \mathbb{C}\), \(C>0\), and let \(D\) be a domain. Let \(\mathcal{F}\) be a family of meromorphic functions in \(D\) such that \(\mathcal{F}_1=\left\lbrace f'/f : f \in \mathcal{F}\right\rbrace \) or \(\mathcal{F}_2=\left\lbrace f''/f : f \in \mathcal{F}\right\rbrace \) is normal in \(D\) and\N\[\N|f''(z)+Af'(z)+Bf(z)|>C \qquad \textrm{for all} \qquad f \in \mathcal{F}, \, z \in D .\N\]\NThen \(\mathcal{F}\) is quasi-normal in \(D\).