Kobayashi length bounds on bordered surfaces and generalized integral points on abelian varieties

Let $B$ be a compact Riemann surface and $B_0\subset B$ a bordered hyperbolic subsurface obtained by removing finitely many disjoint closed disks. Fix a nontrivial loop $\alpha$ in $B_0$. For $s\ge 0$, let $L(\alpha,s)$ denote the supremum, over all finite subsets $S\subset B_0$ with $\#S\le s$, of the minimal Kobayashi length of a loop in $B_0\smallsetminus S$ that is freely homotopic to $\alpha$ in $B_0$. Phung in [7] proved that $L(\alpha,s)$ grows at most linearly and at least as $\sqrt{s}/\log s$. We sharpen the upper bound to $O\left(\sqrt{s\log s}\right)$, which determines $\lim_{s\to\infty}\frac{\log L(\alpha,s)}{\log s}=\frac{1}{2}$, answering a question raised in [7, Question 1.4]. As an application, we improve the counting bound for generalized integral points on abelian varieties over complex function fields: for an abelian variety of dimension $n$ over $\mathbb C(B)$, Phung proved that the number of $(s, B_0)$-generalized integral points modulo the constant trace grows at most as $s^{2nk}$, where $k=\operatorname{rk}(\pi_1(B_0))$. We sharpen this to $s^{nk+\varepsilon}$ for every $\varepsilon>0$, halving the exponent.