A remark about positive polynomials (original) (raw)

Some Extensions of a Theorem of Paul Turán Concerning Polynomials

Kragujevac Journal of Mathematics

In this paper, certain new results concerning the maximum modulus of the polar derivative of a polynomial with restricted zeros are obtained. These estimates strengthen some well known inequalities for polynomial due to Turán, Dubinin and others.

On an inequality of Paul Turan concerning polynomials

Lobachevskii Journal of Mathematics, 2016

Let P (z) be a polynomial of degree n and P s (z) be its sth derivative. In this paper, we shall prove some inequalities for the sth derivative of a polynomial having zeros inside a circle, which as a special case give generalizations and refinements of some results of Turan, Govil, Malik and others.

Some L p Inequalities for Polynomials Not Vanishing inside a Circle

ISRN Mathematical Analysis, 2014

IfP(z)is a polynomial of degreenhaving no zeros in|z|<1, then it is known that, for allα,β∈𝒞with|α|≤1,|β|≤1,R>r≥1, andp>0,P(Rz)-αP(rz)+β{((R+1)/(r+1))n-|α|}P(rz)p≤([Rn-αrn+β{((R+1)/(r+1))n-|α|}]z+[1-α+β{((R+1)/(r+1))n-α}]p/1+zp)P(z)p. In this paper, we will prove a result which not only generalizes the above inequality but also generalize and refines the various results pertaining to theLpnorm ofP(z)∀p>0. We will also prove a result which extends and refines a result of Boas Jr. and Rahman (1962). Also we will see that our results lead to some striking conclusions giving refinements and generalizations of other well-known results.

On polynomials with positive coefficients

Journal of Approximation Theory, 1988

The representation (1) is not unique, since multiplying by we obtain other representations. Among all of the representations (I) of a fixed polynomial p(x) EL, consider those for which m is the least value. This will be called the Lorentz degree of p(x), and it will be by d(p). If ZI" denotes the set of polynomials of degree at most n, then obviously, p(x) E IZ,\ 17, _, implies d(p) 3 Iz.

Some Inequalities for Polynomials Vanishing Inside a Circle

Mathematical journal of interdisciplinary sciences, 2014

In this paper, we will prove a result which not only generalizes the above inequality but also generalize and refines the various results pertaining to the norm of ()∀ > 0. We will also prove a result which extends and refines a result of Boas Jr. and Rahman (1962). Also we will see that our results lead to some striking conclusions giving refinements and generalizations of other well-known results.

Some results concerning the zeros and coefficients of polynomials

МАТЕМАТИЧКИ ВЕСНИК, 2012

Theorem A. Let A(x) = ∑n k=0 akxk, an = 0, be a hyperbolic polynomial with all its zeros x1,x2,...,xn strictly positive. If α, p and b are strictly positive real numbers such that α<p, then n ∑ ... 1 (xp k + b) 1 α ≤ [ α 1 p p 1 α ... Equality holds when A(x) = an ( x − ( bα p − α

A proof of the peak polynomial positivity conjecture

Journal of Combinatorial Theory, Series A, 2017

We say that a permutation π = π 1 π 2 • • • π n ∈ S n has a peak at index i if π i−1 < π i > π i+1. Let P(π) denote the set of indices where π has a peak. Given a set S of positive integers, we define P S (n) = {π ∈ S n : P(π) = S}. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers S and sufficiently large n, |P S (n)| = p S (n)2 n−|S|−1 where p S (x) is a polynomial depending on S. They gave a recursive formula for p S (x) involving an alternating sum, and they conjectured that the coefficients of p S (x) expanded in a binomial coefficient basis centered at max(S) are all nonnegative. In this paper we introduce a new recursive formula for |P S (n)| without alternating sums, and we use this recursion to prove that their conjecture is true.

On roots of polynomials with positive coefficients

Publications de l'Institut Mathematique, 2011

Let α be an algebraic number with no nonnegative conjugates over the field of the rationals. Settling a recent conjecture of Kuba, Dubickas proved that the number α is a root of a polynomial, say P , with positive rational coefficients. We give in this note an upper bound for the degree of P in terms of the discriminant, the degree and the Mahler measure of α; this answers a question of Dubickas.

On some inequalities concerning growth and derivatives of a polynomial

The Journal of Analysis, 2018

A well known result due to Ankeny and Rivlin states that if P(z) = ∑ n =0 a z is a polynomial of degree n with no zeros in |z| < 1 , then for R ≥ 1 , In this paper we obtain some new results which provide several refinements and generalizations of the above inequality and other related results.