Let $A, B$ be $n$-square (Hermitian) positive definite matrices. Let $AB=U|AB|$ be the polar decomposition of $AB$. So $U$ is unitary (called the unitary factor of $AB$). What is the optimal constant $c$ such that $\|I-U\|\le c$, where the norm is the usual spectral norm?

I want to have some understanding on the behaviour of the unitary factor for certain classes of matrices (e.g. matrices with real eigenvalues). Perhaps this is well known, any pointer to the existing papers is welcome.