Analytic knots, satellites and 4-ball genus

Call a knot K in the sphere in complex affine 2-space
analytic if it bounds a complex curve in the ball. The genus
of the curve is the 4-ball genus of the knot. Let K be an
analytic knot. For an arbitrary analytic knot contained in
a small tubular neighbourhood of K we give a sharp lower bound
of the 4-ball genus.
The proof uses mainly complex geometry tools. We do not know
a proof using knot invariants and we do not know whether
the estimate is true for arbitrary quasipositive satellites of
quasipositive knots.