We extend bounds, proved by R.C.~Thompson in 1966, on the sum of the $j$-th largest eigenvalues of the $(n-1) \times (n-1)$ principal matrices of an $n \times n$ Hermitian matrix. Our bounds are stronger than just summing up Thompson's bounds. We achieve the extensions as a corollary of a more general result giving bounds on the zeros of the generalized derivatives of polynomials with real roots. We use the extended bounds to obtain majorization relationships between the eigenvalues of all $m \times m$ principal matrices of an $n \times n$ Hermitian matrix. These majorization relationships imply both a well-known majorization result by Schur and the well-known Szasz's inequalities.