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The generalized Laguerre inequalities and functions in the Laguerre-Pólya class. (English) Zbl 1291.30167

Summary: The principal goal of this paper is to show that the various sufficient conditions for a real entire function, \(\varphi (x)\), 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 \(\varphi (x)\). 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.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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