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 ....
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Published in | Ukrainian mathematical journal Vol. 69; no. 5; pp. 725 - 733 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
New York
Springer US
01.10.2017
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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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 |