I'll need an additional hypothesis: is continuously differentiable. Did you forget it?
Note that if is a linear map, " preserves orthornormal bases" implies for every , (one says that is orthogonal). This implies .
Suppose is defined on an open convex set . By the previous remark, we have for every . As a consequence, for any , the mean-value theorem gives .
In order to get the reverse inequality, prove that, at any point, has locally an inverse function (by the inverse function theorem), notice that this inverse function satisfies the same hypothesis as and procede like above to find if are in a (possibly small) open convex set where is invertible. And this is it.