A submultiplicative property of the Carath\'eodory metric on planar domains

Given a pair of smoothly bounded domains $D_1,D_2\subset \mathbb{C}$, the purpose of this note is to obtain an inequality that relates the Carath'{e}odory metrics on $D_1,D_2,D_1\cap D_2$ and $D_1\cup D_2$.