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 a₀(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 (Formula presented.) 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 (Formula presented.) 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.