A non explicit counterexample to a problem of quasi-normality

In 1986, S.Y. Li and H. Xie proved the following theorem: let k ≥ 2 and let F be a family of functions meromorphic in some domain D , all of whose zeros are of multiplicity at least k . Then F is normal if and only if the family F k = { f ( k ) 1 + | f | k + 1 : f ∈ F } is locally uniformly bounded in D . e give, in the case k = 2 , a counterexample to show that if the condition on the multiplicities of the zeros is omitted, then the local uniform boundedness of F 2 does not even imply quasi-normality. In addition, we give a simpler proof for the Li–Xie theorem (and an extension of it) that does not use Nevanlinna’s Theory which was used in the original proof.