On the order and the lower order of differential polynomials (Q2896926)

Let \(f\) be a transcendental meromorphic function of order \(\sigma(f)\) and of lower order \(\mu(f)\). In this paper, the authors consider the value distribution of a non-constant differential polynomial NEWLINE\[NEWLINE P[f]=\sum_{j=1}^n a_jM_j[f]=\sum_{j=1}^n a_jf^{n_{0,j}}(f')^{n_{1j}}\cdots (f^{(k)})^{n_{kj}}, NEWLINE\]NEWLINE where \(a_j\), \(j=1, 2, \dots, n\), are small functions with respect to \(f\). Define \(d(M_j)=\sum_{i=0}^k n_{ij}\) and denote the degree of \(P\) and the lower degree of \(P\) by \(\overline{d}(P)=\max_{1\leq j\leq n}\{d(M_j)\}\) and \(\underline{d}(P)=\min_{1\leq j\leq n}\{d(M_j)\}\), respectively. It is shown that if NEWLINE\[NEWLINE \sum_{j=1}^n d(M_j)-(n-1)[\overline{d}(P)-\underline{d}(P)]>0, NEWLINE\]NEWLINE then \(\sigma(P)=\sigma(f)\) and \(\mu(P)=\mu(f)\). This result contains some previous results, e.g., [\textit{A. P. Singh}, Indian J. Pure Appl. Math. 16, 791--795 (1985; Zbl 0587.30031)]. The proof uses standard Nevanlinna theory. The authors also obtain a result that holds under the condition \(\overline{N}(r,f)+\overline{N}(r,1/f)=S(r,f)\).