Acta Mathematica Hungarica
Volume/Issue:
Volume 125: Issue 1-2
On the exact Markov inequality for k-monotone polynomials in uniform and L1-norms
Authors:
A. Kroó

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J. Szabados Alfred Rényi Institute of Mathematics 1053 Budapest Reáltanoda u. 13-15. Hungary

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Pages:
99–112
Online Publication Date:
27 Jan 2009
Publication Date:
01 Oct 2009
Article Category:
Research Article
DOI:
https://doi.org/10.1007/s10474-009-8219-6
Keywords:
Markov inequality; k-convex polynomial; Jacobi polynomial; Bessel function; 41A17
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Abstract  

We consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the Markov inequality for monotone polynomials. In this note we give Markov-type inequalities for higher order derivatives in the general class of k-monotone polynomials. In particular, in case of first derivative we find the exact solution of this extremal problem in both uniform and L1-norms. This exact solution is given in terms of the largest zeros of certain Jacobi polynomials.

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Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation		 1950
Volumes
per Year		 3
Issues
per Year		 6
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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