Differential Marty-type inequalities which lead to quasi-normality

We show that a family of all holomorphic functions f in a domain D satisfying 1+|f (m)|alpha (z ) < C, for z is an element of D, m, k is an element of N, m < k, C > 0, is quasi-normal for alpha = 1, |f (k)| and quasi-normal of order at most k - 1 for alpha < 1 (possibly negative) in D. (c) 2023 Elsevier Inc. All rights reserved.