Smooth entrywise positivity preservers, a Horn-Loewner master theorem, and symmetric function identities

A (special case of a) fundamental result of Horn and Loewner [Trans. Amer. Math. Soc. 1969] says that given an integer $n geq 1$, if the entrywise application of a smooth function $f : (0,infty) to mathbb{R}$ preserves the set of $n times n$ positive semidefinite matrices with positive entries, then the first $n$ derivatives of $f$ are non-negative on $(0,infty)$. In a recent joint work with Belton-Guillot-Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and further used it to strengthen the Schoenberg-Rudin characterization of dimension-free positivity preservers [Duke Math. J. 1942, 1959]. In parallel, in recent works with Belton-Guillot-Putinar [Adv. Math. 2016] and with Tao [Amer. J. Math., in press] we used local, real analyt