If $A,Bin{mathcal B}({mathcal H})$ are normal contractions, then for every $Xin {mathcal C}_{left|!!;left|!!;left|cdotright|!!;right|!!;right|}({mathcal H})$ and $alpha > 0$ holdsbegin{equation}bigglvert!bigglvert!bigglvert Bigl(I - A^*ABigr)^{frac{alpha}{2}} X Bigl(I - B^*BBigr)^{frac{alpha}{2}} biggrvert !biggrvert !biggrvert leqslantbigglvert!bigglvert!bigglvert sum_{n=0}^infty (-1)^nbinom{alpha}{n}A^n X B^n biggrvert !biggrvert !biggrvert,end{equation}which generalizes a result of D.R. Joci'c [Proc. Amer. Math. Soc. 126 (1998), no. 9, 2705--2713] for $alpha$ not being an integer. Similar inequalities in the Schatten $p$-norms, for non-normal $A,B$ and in the $Q$-norms if one of $A$ or $B$ is normal, are also given.