Monotonicity for generalized binomial coefficients and Jack positivity

Binomial formulas for Schur polynomials and Jack polynomials were studied by Lascoux in 1978, and Kaneko, Okounkov–Olshanski and Lassalle in the 1990s. We prove that the associated binomial coefficients are monotone and derive some symmetric function inequalities, in particular, a Schur positivity and Jack positivity result. These inequalities are similar to those studied by Newton, Muirhead, Gantmacher, Cuttler–Greene–Skandera, Sra and Khare–Tao.