Inner functions in $C^n$

In this paper we prove that if $f$ is a holomorphic function in a strictly pseudoconvex region $D\subset C^n$, $n>1$, with radial limit equal to 1 in modulus at each point of some nonempty open subset $S$ of the boundary of $D$, then $f\equiv\mathrm{const}$ in $D$.