Abstract. An important chapter in the theory of distribution of zeros of polynomials and transcendental entire functions pertains to the study of linear operators acting on entire functions. This article surveys some recent developments (as well as some classical results) involving some specific classes of linear operators called multiplier sequences and complex zero decreasing sequences. This expository article consists of four parts: Open problems and background information, Composition theorems (Section 2), Multiplier sequences and the Laguerre-Pólya class (Section 3) and Complex zero decreasing sequences (Section 4). A number of open problems and questions are also included.
In this paper we will (1) establish a relationship between the Turan inequalities and the Laguerre inequalities, (2) provide a complete characterization of functions in the Laguerre-Polya class in terms of the Turan inequalities involving the Jensen polynomials and (3) show that certain Hankel determinants of functions in the LaguerrePolya class are nonpositive.
Abstract. The purpose of this paper is to investigate the real sequences γ 0 , γ 1 , γ 2 , . . . with the property that if p(x) = n k=0 a k x k is any real polynomial, then n k=0 γ k a k x k has no more nonreal zeros than p(x). In particular, the authors establish a converse to a classical theorem of Laguerre.
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