  Call a smooth knot (or smooth link) in the unit sphere in $\mathbb{C}^2$
analytic (respectively, smoothly analytic) if it bounds a complex curve
(respectively, a smooth complex curve) in the complex ball. Let $K$ be a
smoothly analytic knot. For a small tubular neighbourhood of $K$ we give a
sharp lower bound for the 4-ball genus of analytic links $L$ contained in it.
