Sharp Deficiency Bounds for Meromorphic Functions in the Unit Disc

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 $0<\lambda (f)\le +\mathrm{\infty }$, and for every positive integer $n$: $$\sum _{a\mathrm{\ne }\mathrm{\infty }}\delta (a,f)\le \delta (0,f^{(n)})(1+nk(f))+\frac{2n(\rho +1)}{\lambda (f)}\mathrm{.}$$ Under the condition $0<\alpha (f)\le +\mathrm{\infty }$, they also obtained $$\sum _{a\mathrm{\ne }\mathrm{\infty }}\delta (a,f)\le \mathrm{\Delta }(0,f^{(n)})(1+nk(f))+\frac{2n}{\alpha (f)}\mathrm{.}$$ Shea and Sons asked whether the factor $2$ could be eliminated. We prove that it can. In fact, whenever $0<\lambda (f)\le +\mathrm{\infty }$, one has $$\sum _{a\mathrm{\ne }\mathrm{\infty }}\delta (a,f)\le \delta (0,f^{(n)})(1+nk(f))+\frac{n(\rho +1)}{\lambda (f)},$$ and under the condition $0<\alpha (f)\le +\mathrm{\infty }$ one has $$\sum _{a\mathrm{\ne }\mathrm{\infty }}\delta (a,f)\le \mathrm{\Delta }(0,f^{(n)})(1+nk(f))+\frac{n}{\alpha (f)}\mathrm{.}$$ The coefficient $n$ in each of $n(\rho +1)\mathrm{/}\lambda (f)$ and $n\mathrm{/}\alpha (f)$ 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 $V$.