AI 中文总结
本文证明单位圆盘内亚纯函数亏量和的Shea-Sons界中因子2可消除，得到最优系数n的锐化界，并推广到微分算子情形。
AI 中文摘要
1986年，Shea和Sons在假设$0<\lambda(f)\leq+\infty$且对每个正整数$n$的条件下，得到了单位圆盘内有限级$\rho$的亚纯函数$f$的如下界：\\[ \sum_{a\ne\infty}\delta(a,f) \leq \delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n(\rho+1)}{\lambda(f)}. \\] 在条件$0<\alpha(f)\leq+\infty$下，他们还得到\\[ \sum_{a\ne\infty}\delta(a,f) \leq \Delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n}{\alpha(f)}. \\] Shea和Sons曾问因子$2$能否被消除。我们证明它可以被消除。事实上，只要$0<\lambda(f)\leq+\infty$，就有\\[ \sum_{a\ne\infty}\delta(a,f) \leq \delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n(\rho+1)}{\lambda(f)}, \\] 且在条件$0<\alpha(f)\leq+\infty$下，有\\[ \sum_{a\ne\infty}\delta(a,f) \leq \Delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n}{\alpha(f)}. \\] 在$n(\rho+1)/\lambda(f)$和$n/\alpha(f)$中的系数$n$对每个$n$都是最优的，且这些项对$\rho$和$\alpha(f)$的依赖也是本质的。最后，我们将这两个估计推广到$n$维复向量空间$V\subset\mathbb{C}(z)$的有限子集$A$，其中$f^{(n)}$被阶为$n$、核为$V$的首一微分算子$D_Vf$取代。我们证明这两个推广对每个$V$都是锐的。
英文摘要
In 1986, Shea and Sons obtained the following bound for a meromorphic function $f$ of finite order $ρ$ in the unit disc, under the hypothesis $0<λ(f)\leq+\infty$, and for every positive integer $n$: \[ \sum_{a\ne\infty}δ(a,f) \leq δ(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n(ρ+1)}{λ(f)}. \] Under the condition $0<α(f)\leq+\infty$, they also obtained \[ \sum_{a\ne\infty}δ(a,f) \leq Δ(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n}{α(f)}. \] Shea and Sons asked whether the factor $2$ could be eliminated. We prove that it can. In fact, whenever $0<λ(f)\leq+\infty$, one has \[ \sum_{a\ne\infty}δ(a,f) \leq δ(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n(ρ+1)}{λ(f)}, \] and under the condition $0<α(f)\leq+\infty$ one has \[ \sum_{a\ne\infty}δ(a,f) \leq Δ(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n}{α(f)}. \] The coefficient $n$ in each of $n(ρ+1)/λ(f)$ and $n/α(f)$ is best possible for every $n$, and the dependence on $ρ$ and $α(f)$ in these terms is also essential. At the end, we extend both estimates to finite subsets $A$ of an $n$-dimensional complex vector space $V\subset\mathbb{C}(z)$, with $f^{(n)}$ replaced by the monic differential operator $D_Vf$ of order $n$ whose kernel is $V$. We prove that both extensions are sharp for every $V$.
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