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 $0<\lambda(f)\leq+\infty$, and for every positive integer $n$: \[ \sum_{a\ne\infty}\delta(a,f) \leq \delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n(\rho+1)}{\lambda(f)}. \] Under the condition $0<\alpha(f)\leq+\infty$, they also obtained \[ \sum_{a\ne\infty}\delta(a,f) \leq \Delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n}{\alpha(f)}. \] Shea and Sons asked whether the factor $2$ could be eliminated. We prove that it can. In fact, whenever $0<\lambda(f)\leq+\infty$, one has \[ \sum_{a\ne\infty}\delta(a,f) \leq \delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n(\rho+1)}{\lambda(f)}, \] and under the condition $0<\alpha(f)\leq+\infty$ one has \[ \sum_{a\ne\infty}\delta(a,f) \leq \Delta(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n}{\alpha(f)}. \] 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 $D_Vf$ of order $n$ whose kernel is $V$. We prove that both extensions are sharp for every $V$.