On the modulus of boundary values of holomorphic functions

A differential geometric method is introduced to study the modulus of boundary values of holomorphic functions on smoothly bounded pseudoconvex domains D in C n , n ⩾ 2 {{\mathbf {C}}^n},n \geqslant 2 . It is shown that functions in A ( D ) A(D) are determined up to a constant factor by their modulus on an open subset of the Shilov boundary. For the case of H ∞ ( D ) {H^\infty }(D) , it is shown that inner functions which satisfy a certain local condition are constant.