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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0269-x,
author = {George Csordas and Anna Vishnyakova},
title = {The generalized Laguerre inequalities and functions in the Laguerre-P\'olya class},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1643-1650},
zbl = {1291.30167},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0269-x}
}
George Csordas; Anna Vishnyakova. The generalized Laguerre inequalities and functions in the Laguerre-Pólya class. Open Mathematics, Tome 11 (2013) pp. 1643-1650. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0269-x/
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