On Row Differential Inequalities Related to Normality and Quasi-normality

We study connections between a new type of linear differential inequalities and normality or quasi-normality. We prove that if C>0 , k≥ 1 and a_0(z),… ,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∈ D | f^(k)(z) + a_k-1(z)f^(k-1)(z)+⋯ +a_0(z)f(z)| < C is quasi-normal in D. For the reversed sign of the inequality we show the following: Suppose that A,B∈ℂ , C>0 and ℱ is a family of meromorphic functions f satisfying for every z∈ D | f^”(z) + Af^'(z) + B f(z)| > C and also at least one of the families { f'/f:f∈ℱ} or { f”/f:f∈ℱ} is normal. Then ℱ is quasi-normal in D.