Given two square matrices $A$ and $B$. There are quite some results on the distance between the eigenvalues, e.g.,

$$ | \lambda_A - \lambda_B | \leq || A - B ||_F, $$ where $A$ and $B$ are Hermitian (see here for more). I am looking for similar results for the argument of the eigenvalues, for example
$$ | arg(\lambda_A) - arg(\lambda_B) | \leq || A - B ||_? $$ One simple instance I have found: if invertible $A$ is normal, given the polar decomposition $A = UP$, then $| arg(\lambda_A) - arg(\lambda_U) | = 0$. Can this result be extended in terms of perturbation?