Exact Constant in Dzyadyk’s Inequality for the Derivative of an Algebraic Polynomial

For natural k and n ≥ 2 k, we determine the exact constant c ( n, k ) in Dzyadyk’s inequality P n ′ φ n 1 − k C − 1 1 ≤ c n k n P n φ n − k C − 1 1 for the derivative P n ′ of an algebraic polynomial P n of degree ≤ n, where φ n x ≔ n − 2 + 1 − x 2 . Namely, c n k = 1 + k 1 + n 2 − 1 n 2 − k ....

Full description

Saved in:
Bibliographic Details
Published inUkrainian mathematical journal Vol. 69; no. 5; pp. 725 - 733
Main Authors Halan, V. D., Shevchuk, I. O.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.10.2017
Springer Nature B.V
Subjects
Algebra
Analysis
Applications of Mathematics
Geometry
Mathematics
Mathematics and Statistics
Statistics
Online AccessGet full text

Cover

More Information
Summary:For natural k and n ≥ 2 k, we determine the exact constant c ( n, k ) in Dzyadyk’s inequality P n ′ φ n 1 − k C − 1 1 ≤ c n k n P n φ n − k C − 1 1 for the derivative P n ′ of an algebraic polynomial P n of degree ≤ n, where φ n x ≔ n − 2 + 1 − x 2 . Namely, c n k = 1 + k 1 + n 2 − 1 n 2 − k .
ISSN:0041-5995
1573-9376
DOI:10.1007/s11253-017-1390-y