WEAKLY INCREASING ZERO-DIMINISHING SEQUENCES

The following problem, suggested by Laguerre's Theorem (1884), remains open: Characterize all real sequences {µ k } ∞ k=0 which have the zero-diminishing property; that is, if p(x) = 1. Introduction. In 1914 Pólya and Schur [14] characterized those linear transformations T of the form

1995

The purpose of this paper is to investigate the real sequences 0, 1, 2,... with the property that if p(x) = n k=0 akx k is any real polynomial, then n k=0 kakx k has no more nonreal zeros than p(x). In particular, the authors establish a converse to a classical theorem of Laguerre.

A complete description is given for the sequences {~'k }~=0 such that, for an arbitrary real polynomial

1997

The problem of characterizing all real sequences 0 with the property that if 0 is any real polynomial, then 0 has no more nonreal zeros than , remains open. Recently, the authors solved this problem under the additional assumption that the sequences 0 , with the aforementioned property, can be interpolated by polynomials. The purpose of this paper is to

Michigan Mathematical Journal, 2001

For z 0 ∈ C and r > 0, let D(z 0 , r) := {z ∈ C : |z − z 0 | < r}. In this paper we show that a polynomial p of the form (*) p(x) = n j=0 a j x j , |a 0 | = 1 , |a j | ≤ 1 , a j ∈ C , has at most (c 1 /α) log(1/α) zeros in the disk D(0, 1 − α) for every α ∈ (0, 1), where c 1 > 0 is an absolute constant. This is a simple consequence of Jensen's formula. However it is not so simple to show that this estimate for the number of zeros in D(0, 1 − α) is sharp. We will present two examples to show the existence of polynomials pα (α ∈ (0, 1)) of the form (*) (with a suitable n ∈ N depending on α) with at least ⌊(c 2 /α) log(1/α)⌋ zeros in D(0, 1 − α) (c 2 > 0 is an absolute constant). In fact, we will show the existence of such polynomials from much smaller classes with more restrictions on the coefficients. Our first example has probabilistic background and shows the existence of polynomials pα (α ∈ (0, 1)) with complex coefficients of modulus exactly 1 and with at least ⌊(c 2 /α) log(1/α)⌋ zeros in D(0, 1 − α) (c 2 > 0 is an absolute constant). Our second example is constructive and defines polynomials pα (α ∈ (0, 1)) with real coefficients of modulus at most 1, with constant term 1, and with at least ⌊(c 2 /α) log(1/α)⌋ zeros in D(0, 1 − α) (c 2 > 0 is an absolute constant).

Proceedings of the American Mathematical Society, 2008

We describe all zero-diminishing sequences (over the real-valued polynomials on R) which additionally satisfy a Carleman condition and show that they are of the same kind as those in E. Laguerre's theorem from 1884.

2019

This licentiate consists of two papers treating polynomial sequences defined by linear recurrences.In paper I, we establish necessary and sufficient conditions for the reality of all the zeros in a ...

Journal of Approximation Theory, 2015

The purpose of this note is to provide sufficient geometric conditions on E under which the (full) sequence of normalized counting measures of the zeros of {P n } converges in the weak-star topology to the equilibrium measure on E, as n → ∞. Utilizing an argument of Gardiner and Pommerenke dealing with the balayage of measures, we show that this is true, for example, if the interior of the polynomial convex hull of E has a single component and the boundary of this component has an "inward corner" (more generally, a "non-convex singularity"). This simple fact has thus far not been sufficiently emphasized in the literature. As applications we mention improvements of some known results on the distribution of zeros of some special polynomial sequences. Dedication: To Herbert Stahl, an exceptional mathematician, a delightful personality, and a dear friend.

Czechoslovak Mathematical Journal, 2020

It is known that a set H of positive integers is a Poincaré set (also called intersective set, see I. Ruzsa (1982)) if and only if dim H (X H) = 0, where X H := x = ∞ n=1 xn 2 n : xn ∈ {0, 1}, xnx n+h = 0 for all n 1, h ∈ H and dim H denotes the Hausdorff dimension (see C. Bishop, Y. Peres (2017), Theorem 2.5.5). In this paper we study the set X H by replacing 2 with b > 2. It is surprising that there are some new phenomena to be worthy of studying. We study them and give several examples to explain our results.

Journal of Combinatorial Theory, Series A, 2005

Let f (x) and g(x) be two real polynomials whose leading coefficients have the same sign. Suppose that f (x) and g(x) have only real zeros and that g interlaces f or g alternates left of f . We show that if ad ≥ bc then the polynomial