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Value distribution of certain differential polynomials. (English) Zbl 0999.30023
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 $\begin{split}(n- \gamma_{Q_2}) T(r,f)\leq\overline N(r,0; F)+\overline N(r,0; P[f])+\\ (\Gamma_{Q_2}- \gamma_{Q_2}+ 1)\overline N(r,f)- \gamma\{N(r,0; f)- N_{k+1}(r, 0;f)\}+ S(r,f),\end{split}$ 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 H. X. Yi [J. Math. Anal. Appl. 154, No. 2, 318-328 (1991; Zbl 0725.30022)].

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Zbl 0725.30022
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