George Csordas

Composition Theorems, Multiplier Sequences and Complex Zero Decreasing Sequences

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-Polya class (Section 3) and Complex zero decreasing sequences (Section 4). A number of open problems and questions are also included.

Comparisons of regular splittings of matrices

SummaryIn this article, new comparison theorems for regular splittings of matrices are derived. In so doing, the initial results of Varga in 1960 on regular splittings of matrices, and the subsequent unpublished results of Woźnicki in 1973 on regular splittings of matrices, will be seen to be special cases of these new comparison theorems.

The Riemann hypothesis and the Turán inequalities

A solution is given to a fifty-eight year-old open problem of G Polya, involving the Taylor coefficients of the Riemann ?-function. 1. Introduction. The purpose of this paper is to solve a fifty-eight year-old problem of Polya (P, p. 16), related to the Riemann Hypothesis. This problem may be described as follows. Starting with Riemanns definition of his ?-function (cf. Titchmarsh (T, p. 16), in a slightly different notation), i.e.,

Necessary and sufficient conditions and the Riemann hypothesis

The purpose of this paper is (1) to investigate several necessary and sufficient conditions for a real entire function to have only real zeros and to apply these conditions to the Riemann f) := 8/Dm@( t)cos( xt) dt are real, where and where a(t) := 2 a,(t)

Moment inequalities and the Riemann hypothesis

AbstractIt is known that the Riemann hypothesis is equivalent to the statement that all zeros of the Riemann ξ-function are real. On writingξ(x/2)=8 ∫0∞ Φ(t) cos(xt)dt, it is known that a necessary condition that the Riemann hypothesis be valid is that the moments

A low bound for the de Bruijn-newman constant Λ

\hat b_m (\lambda ): = \int_0^\infty {t^{2m} e^{\lambda t^2 } \Phi (t)dt}

A new property of a class of Jacobi polynomials

satisfy the Turán inequalities(*)