Log-convexity properties of Schur functions and generalized hypergeometric functions of matrix argument

Abstract

We establish a positivity property for the difference of products of certain Schur functions, s λ (x), where λ varies over a fundamental Weyl chamber in R n and x belongs to the positive orthant in R n . Further, we generalize that result to the difference of certain products of arbitrary numbers of Schur functions. We also derive a log-convexity property of the generalized hypergeometric functions of two Hermitian matrix arguments, and we show how that result may be extended to derive higher-order log-convexity properties.

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“…Although we will not need this extension, it does give a more conceptual proof of Björner's inequality. In a different direction, Richards [34] gave an analytic generalization of (4-1) for real λ, µ ∈ ‫ޒ‬ ℓ and the determinant definition of Schur polynomials. It would be natural to conjecture that (4-4) also generalizes to this setting.…”