Abstract
Applying the factorization of some related matrix functions, we investigate some inverse problems with mixed spectral data for Jacobi matrices and Stieltjes strings. Besides, we prove a discrete analogue of Borg's theorem for the Green's matrix. We also study the first eigenvalue, and the ratio of the first two eigenvalues of the Stieltjes string equation. With certain restrictions on the class of density sequences $p$, we determine the shapes of the extremal density sequence for the first eigenvalue, and the minimum for the ratio of the first two eigenvalues.