Jensen polynomials and the Turán and Laguerre inequalities

Craven, Thomas C.; Csordás, George

doi:10.2140/pjm.1989.136.241

Cited by 80 publications

(76 citation statements)

References 10 publications

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“…A univariate polynomial with real coefficients is stable if and only if it has all real zeros, so the familiar Newton inequalities (cf., e.g., [22,38]) imply that f satisfies the conclusion of Conjecture 2.4 if f is stable with non-negative coefficients.…”

Section: Conjecture 29 Suppose That µ Is Cna+ and Thatmentioning

confidence: 99%

“…The well-known Laguerre-Turán inequalities for univariate polynomials and transcendental entire functions in the Laguerre-Pólya class [22] amount to saying that the sequence of Taylor coefficients of such a function is SLC (Definition 2.8). One can argue that both Newton's inequalities and the Laguerre-Turán inequalities are natural manifestations of negative dependence properties encoded in the geometry of the zero sets of these functions.…”

Section: Conjecture 29 Suppose That µ Is Cna+ and Thatmentioning

confidence: 99%

“…The first inequality comes from the already proved part of (22), while the second is obtained by repeated application of (23). Then µ is CNA.…”

Section: Then µ Is Rayleigh If and Only Ifmentioning

confidence: 99%

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Negative dependence and the geometry of polynomials

Borcea

Brändén

Liggett

2008

J. Amer. Math. Soc.

204

312

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We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures and uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence, and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons’ recent results on determinantal measures, and we construct counterexamples to several conjectures of Pemantle and Wagner on negative dependence and ultra log-concave rank sequences.

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Section: Conjecture 29 Suppose That µ Is Cna+ and Thatmentioning

confidence: 99%

Section: Conjecture 29 Suppose That µ Is Cna+ and Thatmentioning

confidence: 99%

See 1 more Smart Citation

Negative dependence and the geometry of polynomials

Borcea

Brändén

Liggett

2008

J. Amer. Math. Soc.

204

312

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“…We have adopted the notations in [2,4,14], which one may consult for the important properties of the functions in Laguerre-Pólya class. Generally, L-P consists of entire functions which are uniform limits on the compact sets of the complex plane of real polynomials with only real zeros.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Higher order Turán inequalities

Dimitrov

1998

Proc. Amer. Math. Soc.

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Abstract. The celebrated Turán inequalitieswhere Pn(x) denotes the Legendre polynomial of degree n, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities γ 2 n − γ n−1 γ n+1 ≥ 0, n ≥ 1, which hold for the Maclaurin coefficients of the real entire function ψ in the Laguerre-Pólya class, ψ(x) = ∞ n=0 γnx n /n!.

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“…In this setting, such (and analogous) inequalities are commonly known as Turán Inequalities (Turán-like Inequalities). Concerning Turán inequalities see, for example, [KaSz] and [CrCs2].…”

Section: Properties Of Steiner Polynomialsmentioning

confidence: 99%

On H. Weyl and J. Steiner Polynomials

Katsnelson

2008

Complex Anal. Oper. Theory

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Abstract. The paper deals with root location problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set V ⊂ R n . A polynomial of this class describes the volume of the set V + tB n as a function of t, where t is a positive number and B n denotes the unit ball in R n . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold M, where M is isometrically embedded with positive codimension in R n . A Weyl polynomial describes the volume of a tubular neighborhood of its associated M as a function of the tube's radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of R n into R m for m > n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial's roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold.

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Jensen polynomials and the Turán and Laguerre inequalities

Cited by 80 publications

References 10 publications

Negative dependence and the geometry of polynomials

Negative dependence and the geometry of polynomials

Higher order Turán inequalities

On H. Weyl and J. Steiner Polynomials

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