On the existence of independent inner functions

The existence of independent inner functions is established: there exist inner functions $\phi$ and $\psi$, both vanishing at the origin, such that $$\frac{1}{2\pi }{\int }_0^{2\pi }{\phi }^m(e^{i\theta })\overline{{\psi }^n(e^{i\theta })}d\theta =0,m,n\ge 1.$$ In fact, both $\phi$ and $\psi$ can be chosen to be Blaschke products. The proof proceeds by an iterative construction, which suggests a general approach to produce inner functions with a desired joint distribution.