Let $C_1,\ldots,C_n$ by simply-connected polygons in the plane. Riemann's mapping theorem implies that, for each individual polygon $C_i$, there is an analytic bijection $f_i$ that maps $C_i$ to any given convex polygon (e.g. the unit square).
Is there a single analytic bijection $f$ on the plane, that maps each $C_i$ to a convex polygon?