Markov-Bernstein-type inequalities for classes of polynomials with restricted zeros
نویسندگان
چکیده
منابع مشابه
Markov-Bernstein Type Inequalities for Classes of Polynomials with Restricted Zeros
We prove that there exists an absolute constant c > 0 such that |p′(y)| ≤ c min { n(k + 1), ( n(k + 1) 1 − y2 )1/2} max −1≤x≤1 |p(x)|, −1 ≤ y ≤ 1 for every real algebraic polynomial of degree at most n having at most k zeros in the open unit disk {z ∈ C : |z| < 1}. This inequality, which has been conjectured for at least a decade, improves and generalizes several earlier results. Up to the mult...
متن کاملMarkov- and Bernstein-type Inequalities for Polynomials with Restricted Coefficients
The Markov-type inequality ‖p′‖[0,1] ≤ cn log(n + 1)‖p‖[0,1] is proved for all polynomials of degree at most n with coefficients from {−1, 0, 1} with an absolute constant c. Here ‖·‖[0,1] denotes the supremum norm on [0, 1]. The Bernstein-type inequality |p′(y)| ≤ c (1 − y)2 ‖p‖[0,1] , y ∈ [0, 1) , is shown for every polynomial p of the form
متن کاملMARKOV AND BERNSTEIN TYPE INEQUALITIES IN Lp FOR CLASSES OF POLYNOMIALS WITH CONSTRAINTS
The Markov-type inequality i \Ax)\dx is proved for all real algebraic polynomials/of degree at most n having at most k, with 0 ^ k < n, zeros (counting multiplicities) in the open unit disk of the complex plane, and for all p > 0, where c(p) = c(l +/T) with some absolute constant c> 0. This inequality has been conjectured since 1983 when the L^ case of the above result was proved. It improves a...
متن کاملSome compact generalization of inequalities for polynomials with prescribed zeros
Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$. In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$, $k^2 leq rRleq R^2$ and for $Rleq r leq k$. Our results refine and generalize certain well-known polynomial inequalities.
متن کاملMarkov-bernstein Type Inequalities for Polynomials under Erdős-type Constraints
Throughout his life Erdős showed a particular fascination with inequalities for constrained polynomials. One of his favorite type of polynomial inequalities was Markovand Bernstein-type inequalities. For Erdős, Markovand Bernstein-type inequalities had their own intrinsic interest. He liked to see what happened when the polynomials are restricted in certain ways. Markovand Bernstein-type inequa...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Constructive Approximation
سال: 1994
ISSN: 0176-4276,1432-0940
DOI: 10.1007/bf01212567