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The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

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Csordas G., Vishnyakova A.

Open Mathematics

2013

tom 11

nr 9

1643-1650

The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.

The cone of nonnegative polynomials with nonnegative coefficients and linear operators preserving this cone

100%

Katkova O., Vishnyakova A.

Open Mathematics

2014

tom 12

nr 5

752-760

Let be the cone of real univariate polynomials of degree ≤ 2n which are nonnegative on the real axis and have nonnegative coefficients. We describe the extremal rays of this convex cone and the class of linear operators, acting diagonally in the standard monomial basis, preserving this cone.

Ograniczanie wyników

2 Open Mathematics

2 Vishnyakova A.

1 Csordas G.

1 Katkova O.

1 2014

1 2013

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The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

The cone of nonnegative polynomials with nonnegative coefficients and linear operators preserving this cone