On Row Differential Inequalities Related to Normality and Quasi-normality

Abstract

We study connections between a new type of linear differential inequalities and normality or quasi-normality. We prove that if \(C>0\), \(k\ge 1\) and \(a_0(z),\dots ,a_{k-1}(z)\) are fixed holomorphic functions in a domain D, then the family of the holomorphic functions f in D, satisfying for every \(z\in D\)

$$\begin{aligned} \left| f^{(k)}(z) + a_{k-1}(z)f^{(k-1)}(z)+\cdots +a_0(z)f(z)\right| < C \end{aligned}$$

is quasi-normal in D. For the reversed sign of the inequality we show the following: Suppose that \(A,B\in {{\mathbb {C}}}\), \(C>0\) and \(\mathcal {F}\) is a family of meromorphic functions f satisfying for every \(z\in D\)

$$\begin{aligned} \left| f^{''}(z) + Af^{'}(z) + B f(z)\right| > C \end{aligned}$$

and also at least one of the families \(\left\{ f'/f:f\in \mathcal {F}\right\} \) or \(\left\{ f''/f:f\in \mathcal {F}\right\} \) is normal. Then \(\mathcal {F}\) is quasi-normal in D.