The purpose of this paper is (1) to highlight some recent and heretofore
unpublished results in the theory of multiplier sequences and (2) to survey
some open problems in this area of research. For the sake of clarity of exposition,
we have grouped the problems in three subsections, although several of the problems
are interrelated. For the reader’s convenience, we have included the pertinent
definitions, cited references and related results, and in several instances, elucidated
the problems by...

The following problem, suggested by Laguerre’s Theorem (1884),
remains open: Characterize all real sequences {μk} k=0...∞
which have the zero-diminishing property; that is, if k=0...n, p(x) = ∑(ak x^k) is any P real polynomial, then
k=0...n, p(x) = ∑(μk ak x^k) has no more real zeros than p(x).
In this paper this problem is solved under the additional assumption of a weak
growth condition on the sequence {μk} k=0...∞, namely lim n→∞ | μn |^(1/n) < ∞.
More precisely, it is established that...

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...

The Laguerre inequality and the distribution of zeros of real entire functions are investigated with the aid of certain infinite-order differential operators. The paper includes new proofs, problems, conjectures and many illustrative examples and counterexamples.

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