Let $H$ be a Hermitian $n \times n$ matrix. Let $V$ be another such matrix.
For real $t$, let us consider the one-parameter family 
$$ H(t) = H + t V$$
of Hermitian matrices.
[Kato's](https://www.springer.com/de/book/9783540586616) perturbation theory tells us that the eigenvalues $\lambda_k(t)$ and eigenfunctons $\phi_k(t)$ of this matrix-family can be chosen to beanalytic in $t$ and there is a family of unitary matrices $U(t)$ so that $\phi_k(t) = U(t)\phi_k(0)$.

Are there constants so that 
$$|\lambda_k(1)-\lambda_k(0)| \leq C_k ||V|| $$
$$||U(1)-U(0)|| \leq C ||V|| $$
holds true without assuming that all eigenvalues are simple?
What are those constants?