We prove a monotonicity phenomenon for ratios of Schur polynomials. In this we are motivated by - and apply our result to - understanding polynomials and power series that preserve positive semidefiniteness (psd) when applied entrywise to psd matrices. We then extend these results to classify polynomial preservers of total positivity. As a further application, we extend a conjecture of Cuttler, Greene, and Skandera (2011) to obtain a novel characterization of weak majorization using Schur polynomials. Our proofs proceed through a Schur positivity result of Lam, Postnikov, and Pylyavskyy (2007), and computing the leading terms of Schur polynomials.