Suppose z= phi(&) and w=psi(&) are one-to-one analytic maps from the unit disc D(0,1) onto the regions G_1 and G_2. Set phi(0)=z_0 and psi(0)=w_0. Let 0<r<1 and
omega_1(r)=phi(D(0,r)), omega_2(r)=psi(D(0,r)). Assume f: G_1->G-2 be holomorphic map with f(z_0)=w_0. Show that f(omega_1(r)) is contained in omega_2(r)
This provides an example of working with a holomorphic map.