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

• Autorzy: George Csordas , Anna Vishnyakova
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Laguerre-Pólya class; Generalized Laguerre-type inequalities

Kategorie tematyczne

• 30D35: Distribution of values, Nevanlinna theory
• 30D15: Special classes of entire functions and growth estimates

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 9
• Strony: 1643-1650

Daty

wydano

2013-09-01

online

2013-06-28

Twórcy

autor

George Csordas

• University of Hawaii

autor

Anna Vishnyakova

• Kharkov National University

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-013-0269-x