Weakly increasing zero-diminishing sequences

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

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

Creative Mathematics and Informatics, 2016

In this paper we consider for a fixed µ, the class of polynomials P (z) = a 0 + n ν=µ aν zν , 1 ≤ µ ≤ n, of degree at most n not vanishing in the disk |z| < k, k > 0. For any ρ > σ ≥ 1 and 0 < r ≤ R ≤ k, we investigate the dependence of P (ρz) − P (σz) R on P r and derive various refinements and generalizations of some well known results.

In the present paper we consider $F_k(x)=x^{k}-\sum_{t=0}^{k-1}x^t,$ the characteristic polynomial of the $k$-th order Fibonacci sequence, the latter denoted $G(k,l).$ We determine the limits of the real roots of certain odd and even degree polynomials related to the derivatives and integrals of $F_k(x),$ that form infinite sequences of polynomials, of increasing degree. In particular, as $k \to \infty,$ the limiting values of the zeros are determined, for both odd and even cases. It is also shown, in both cases, that the convergence is monotone for sufficiently large degree. We give an upper bound for the modulus of the complex zeros of the polynomials for each sequence. This gives a general solution related to problems considered by Dubeau 1989, 1993, Miles 1960, Flores 1967, Miller 1971 and later by the second author in the present paper, and Narayan 1997.

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

2012

In this paper we study the growth of polynomials of degree n having all its zeros on |z| = k, k ≤ 1 using the notation M(p, t) = max |z|=t |p(z)|, we measure the growth of p by estimating { M(p,t) M(p,1) }s from above for any t ≥ 1, s being an arbitrary positive integer. DOI: 10.1134/S1995080212020102

Ukrainian Mathematical Journal, 1992

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.

Pacific Journal of Mathematics, 1981

An inequality is established which provides a unifying principle for the distribution of zeros of real polynomials and certain entire functions. This inequality extends the applicability of multiplier sequences to the class of all real polynomials. The various consequences obtained generalize and supplement several results due to Hermite-Poulain, Laguerre, Marden, Obreschkoff, Polya and Schur.