Value distribution of certain differential polynomials
Let \(f\) be a transcendental meromorphic function in \(\mathbb{C}\). The author considers the value distribution of differential polynomials. Let \(n_{0j}, n_{1j},\dots, n_{kj}\) be nonnegative integers. For a differential monomial \(M_j[f]= (f)^{n_{0j}}(f')^{n_{1j}} (f'')^{n_{2j}}\cdots (f^{(k)})^{n_{kj}}\), we define a degree \(\gamma_M= \sum^k_{i=0} n_{ij}\) and a weight \(\Gamma_{M_j}= \sum^k_{i=0} (i+ 1)n_{ij}\). The sum \(P[f]= \sum^l_{i=1} b_j M_j[f]\) is called a differential polynomial in \(f\) of degree \(\gamma_P= \max\{\gamma_{M_j}: 1\leq j\leq l\}\) and of weight \(\Gamma_P= \max\{\Gamma_{M_j}: 1\leq j\leq l\}\), where \(T(r, b_j)= S(r,f)\) for \(j= 1,2,\dots, l\). The numbers \(\underline{\gamma_P}= \min\{\gamma_{M_j}: 1\leq j\leq l\}\) and \(k\) are called a lower degree and an order of \(P[f]\), respectively. The main result in this paper is the following: let \(Q_1[f](\not\equiv 0)\), \(Q_2[f](\not\equiv 0)\) be two differential polynomials in \(f\) such that \(k\) be the order of \(Q_1[f]\), and let \(P[f]= \sum^n_{i=0} a_i f^i\), where \(a_n(\not\equiv 0)\), \(a_{n-1},\dots, a_0\) are small functions with respect to \(f\). For a differential polynomial \(F= P[f]Q_1[f]+ Q_2[f]\), we have \[ \begin{multlined}(n- \gamma_{Q_2}) T(r,f)\leq\overline N(r,0; F)+\overline N(r,0; P[f])+\\ (\Gamma_{Q_2}- \gamma_{Q_2}+ 1)\overline N(r,f)- \gamma\{N(r,0; f)- N_{k+1}(r, 0;f)\}+ S(r,f),\end{multlined} \] where \(\gamma= \underline{\gamma_{Q_1}}\) if \(n\geq \gamma_{Q_2}\) and \(\gamma= 0\) if \(n< \gamma_{Q_2}\). This result is an improvement of \textit{H. X. Yi} [J. Math. Anal. Appl. 154, No. 2, 318-328 (1991; Zbl 0725.30022)].
scientific article; zbMATH DE number 1751289