On Sharpening of a Theorem of Ankeny and Rivlin

Let $p(z)=\sum^n_{v=0}a_vz^v$be a polynomial of degree $n$, $M(p,R)=:\underset{|z|=R\geq 0}{\max}|p(z)|$ and $M(p,1)=:||p||$.Then according to a well-known result of Ankeny and Rivlin [1], we have for $R\geq 1$, $$M(p,R)\leq (\frac{R^n+1}{2})||p||.$$This inequality has been sharpened by Govil [4], who proved that for $R\geq 1$, $$M(p,R)\leq (\frac{R^n+1}{2})||p||-\frac{n}{2}(\frac{||p||^2-4|a_n|^2}{||p||})\left\{\frac{(R-1||p||)}{||p||+2|a_n|}-ln(1+\frac{(R-1)||p||}{||p||+2|a_n|})\right\}.$$In this paper, we sharpen the above inequality of Govil [4], which in turn sharpens the inequality of Ankeny and Rivlin [1].     

Bounds on the moduli of polynomial zeros

In this paper, we develop methods for establishing improved bounds on the moduli of the zeros of complex and real polynomials. Specific (lacunary) as well as arbitrary polynomials are considered. The methods are applied to specific polynomials by way of example. Finally, we evaluate the quality of some bounds numerically.     

关于T.J.Rivlin的一个问题

本文在连续函数空间内按两种范数‖·‖_Ｍ（Ｏｒｌｉｃｚ范数）和‖·‖_(Ｍ)（Ｌｕｘｅｍｂｕｒｇ范数分别解决了Ｔ．Ｊ．Ｒｉｖｌｉｎ的一个问题     

Inequalities for a polynomial and its derivative

For an arbitrary entire functionf and anyr>0, letM(f,r):=max_|z|=r |f(z)|. It is known that ifp is a polynomial of degreen having no zeros in the open unit disc, andm:=min_|z |=1|p(z)|, then

On a quadrature formula of Micchelli and Rivlin

Micchelli and Rivlin (1972) obtained a quadrature formula of highest algebraic degree of precision for the Fourier-Chebyshev coefficients A_n(f), which is based on the divided differences of f′ at the zeros of the Chebyshev polynomial T_n(x). We give here a simple approach to questions of this type, which applies to the coefficients in arbitrary orthogonal expansion of f. As an auxiliary result we obtain a new interpolation formula and a new representation of the Turán quadrature formula.     

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.

     

The expected value of the number of real zeros of a random sum of Legendre polynomials

It is known that the expected number of zeros in the interval of the sum , in which is the normalized Legendre polynomial of degree and the coefficients are independent normally distributed random variables with mean 0 and variance 1, is asymptotic to for large . We improve this result and show that this expected number is for any positive .

     

On a conjectured inequality of Gautschi and Leopardi for Jacobi polynomials

Motivated by work on positive cubature formulae over the spherical surface, Gautschi and Leopardi conjectured that the inequality holds for α,β > − 1 and n ≥ 1, θ ∈ (0, π), where are the Jacobi polynomials of degree n and parameters (α, β). We settle this conjecture in the special cases where .      

Generalization of certain well-known polynomial inequalities

If is a polynomial of degree n having no zeros in |z|<1, then for |β|?1, it was proved by Jain [V.K. Jain, Generalization of certain well known inequalities for polynomials, Glas. Mat. 32 (52) (1997) 45-51] that

     

A new property of a class of Jacobi polynomials

Polynomials whose coefficients are successive derivatives of a class of Jacobi polynomials evaluated at are stable. This yields a novel and short proof of the known result that the Bessel polynomials are stable polynomials. Stability-preserving linear operators are discussed. The paper concludes with three open problems involving the distribution of zeros of polynomials.

     

On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Corresponding Orthogonal Polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U U=J _,+A ₁(x–1)+B ₁(x+1)–A ₂(x–1)–B ₂(x+1), where J _, is the Jacobi linear functional, i.e. J _,,p›=_–1 ¹ p(x)(1–x)(1+x)dx,,>–1, pP, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (–1,1) (inner asymptotics) and C[–1,1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A ₂=B ₂=0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n–1/n] Padé approximants are our orthogonal polynomials.     

Trigonometric polynomials with many real zeros and a Littlewood-type problem

We examine the size of a real trigonometric polynomial of degree at most having at least zeros in (counting multiplicities). This result is then used to give a new proof of a theorem of Littlewood concerning flatness of unimodular trigonometric polynomials. Our proof is shorter and simpler than Littlewood's. Moreover our constant is explicit in contrast to Littlewood's approach, which is indirect.

     

Conjectured inequalities for Jacobi polynomials and their largest zeros

Inequalities are conjectured for the Jacobi polynomials and their largest zeros. Special attention is given to the cases β = α − 1 and β = α.      

On conjectured inequalities for zeros of Jacobi polynomials

Inequalities for the largest zero of Jacobi polynomials, conjectured recently by us and in joint work with P. Leopardi, are here extended to all zeros of Jacobi polynomials, and new relevant conjectures are formulated based on extensive computation.      

Some refinements of Enestrom-Kakeya theorem

In this paper we present certain interesting refinements of a well-known Enestrom-Kakeya theorem in the theory of distribution of zeros of polynomials which among other things also improve upon some results of Aziz and Mohammad, Govil and Rehman and others.     

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.Research of Juan José Moreno Balcázar was partially supported by Ministerio de Educación y Ciencia of Spain under grant MTM2005-08648-C02-01 and Junta de Andalucía (FQM 229 and FQM 481).     

Polynomials associated with a functional product of the Hermite type

We have found the motivation for this paper in the research of a quantized closed Friedmann cosmological model. There, the second‐order linear ordinary differential equation emerges as a wave equation for the physical state functions. Studying the polynomial solutions of this equation, we define a new functional product in the space of real polynomials. This product includes the indexed weight functions which depend on the degrees of participating polynomials. Although it does not have all of the properties of an inner product, a unique sequence of polynomials can be associated with it by an additional condition. In the special case presented here, we consider the Hermite‐type weight functions and prove that the associated polynomial sequence can be expressed in the closed form via the Hermite polynomials. Also, we find their Rodrigues‐type formula and a four‐term recurrence relation. In contrast to the zeros of Hermite polynomials, which are symmetrically located with respect to the origin, the zeros of the new polynomial sequence are all positive. Copyright © 2015 John Wiley & Sons, Ltd.