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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.

Ograniczanie wyników

1 Open Mathematics

1 Csordas G.

1 Vishnyakova A.

1 2013

Wyniki wyszukiwania

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