Sharp Deficiency Bounds for Meromorphic Functions in the Unit Disc

Abstract

In 1986, Shea and Sons obtained the following bound for a meromorphic function $f$ of finite order $\rho$ in the unit disc, under the hypothesis $0<\lambda (f)\le +\infty$, 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 $0<\alpha (f)\le +\infty$, 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 $0<\lambda (f)\le +\infty$, 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 $0<\alpha (f)\le +\infty$ 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 $n(\rho +1)/\lambda (f)$ and $n/\alpha (f)$ is best possible for every $n$, and the dependence on $\rho$ and $\alpha (f)$ 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 $D_{V}f$ of order $n$ whose kernel is $V$. We prove that both extensions are sharp for every $V$.

arXiv comment27 pages