Further results about normal criteria and shared values for families of meromorphic functions

Abstract

. Let k be a positive integer and let F be a family of meromorphic functions in the domain D all of whose zeros with multiplicity at least k . Let P be a polynomial and P have at least one simple zero, p = deg ( P ) (cid:2) k + 2. If, for each pair f , g ∈ F , P ( f ) G m ( f ) and P ( g ) G m ( g ) share a nonzero constant b ignoring multiplicity in D, where G ( f ) = P ( f ( k ) )+ H ( f ) is a differential polynomial of f satisfying w deg | H (cid:3) kmql + mq + 1 or w ( H ) − deg ( H ) < qk , and q > l (cid:2) k + 1 is a positive integer, then F is normal in D.