Abstract: By using Montel’s criterion and combining it with relevant theories  of differential equations, we studied the quasi-normal criterion and the order of quasi-normality for a class of meromorphic functions satisfying third-order linear differential inequalities with constant coefficients. The results show  that for all f∈F, when |f(z)+Af(z)|>Cand  F₁={f′(z)/f(z)|f(z)∈F} or F₂={f(z)/f(z)|f(z)∈F is normal,  the family F of meromorphic functions is quasi-normal.

Key words:  , meromorphic function, quasi-normal family, differential inequality, normal criterion