Let $T = U|T|$ be a bounded linear operator with the associated polar decomposition on a separable infinite dimensional Hilbert space. For $0 < t < 1$ , let $T_t = |T|^tU|T|^{1-t}$ and $g_T$ and $g_{T_t}$ be the principal functions of $T$ and $T_t$ , respectively. We show that, if $T$ is an invertible semi-hyponormal operator with trace class commutator $[|T|,U]$ , then $g_T = g_{T_t}$ almost everywhere on $\bm{C}$ . As a biproduct we reprove Berger's theorem and index properties of invertible $p$ -hyponormal operators.