Some results on normal family of meromorphic functions (Q5952803)

The authors present two theorems in normal families. The main result concerns a family \({\mathcal F}\) of meromorphic functions in the unit disk \(D\). Let \(k\) and \(q\) be integers which are at least 2, and for \(f\) in \({\mathcal F}\), let \(H(f,f', \dots, f^{(k)})\) be a differental polynomial of the form \[ \sum^m_{j=1} a_jM_j(f,f', \dots,f^{(k)}) \] where the \(a_j\) are analytic functions in \(D\) and the \(M_j\) are differential monomials in \(f\) with degree \(\gamma_{M_j}\) and weight \(\Gamma_{M_j}\). If \(\max\{\Gamma_{M_j}/ \gamma_{M_j}\} <k+1\), the zeros of \(f\) are of multiplicity \(\leq k+1\), and \((f^{(k)})^q +H(f,f', \dots, f^{(k)})\neq 1\) for each \(f\) in \({\mathcal F}\), then \({\mathcal F}\) is normal in \(D\). It should be noted that \(H\) need not be a homogeneous differential polynomial.