Symmetric polynomials , p - norm inequalities , and certain functionals related to majorization .

We study the relation x ≺L y on [0,∞)n defined by x ≺L y ⇔ ∑n i=1 ψ(xi) ≤ ∑n i=1 ψ(yi) for all ψ : [0,∞) → [0,∞) of the form ψ(s) = ∫ s 0 φ(t) dt t where φ is concave nondecreasing. (We also briefly explain how this arises in the context of some Lp inequalities between complex exponential sums conjectured by Hardy and Littlewood, and why the more familiar relation obtained by allowing any concave nondecreasing ψ (a version of weak majorization) does not hold in that context.) We attempt to characterize x ≺L y by means of another relation, x ≺F y, defined by Fk,r(x) ≤ Fk,r(y), ∀ k, r ≥ 1, where Fk,r(x) is the coefficient of t k in ∏n i=1 ( 1 + (xit) 1/1! + · · · + (xit)/r! ) . We prove that x ≺F y ⇒ x ≺L y. Regarding the converse, we prove a necessary and sufficient condition in terms of ∇g for a function g to have the order-preserving property x ≺L y ⇒ g(x) ≤ g(y) “locally”, and we verify that the condition holds for all g = Fk,r. We then propose a general conjecture that the total positivity of certain Jacobian matrices implies the “path connectedness” of relations such as x ≺L y. If true, this conjecture would allow us to remove the word “locally” and thus complete the proof of x ≺L y ⇒ x ≺F y. Research supported by NSERC Canada. A.M.S. Mathematics Subject Classification : 52A40 (15A42, 15A48, 26B25, 42A05, 47A30, 60E15).