Some analogues of von-Heumann's inequality for $J$-contractions

Let $J$ be a self-adjoint operator satisfying $J^2=I$. We prove that for any $J$-contraction $T$ (i. e. $T^*JT-J\leqslant0$) and any inner function $f$ holomorphic on the spectrum of $T$ the function $f(T)$ is a $J$-contraction too. It is also proved that for $J\ne\pm I$ only inner functions $f$ satisfy this property. We consider other analogues of von-Neumann's inequality.