Markov inequality

for derivatives of algebraic polynomials

An equality giving an estimate of the uniform norm of the derivative in terms of the uniform norm of the polynomial itself. Let $ P _ {n} ( x) $ be an algebraic polynomial of degree not exceeding $ n $ and let

$$ M = \max _ {a \leq x \leq b } | P _ {n} ( x) | . $$

Then for any $ x $ in $ [ a , b ] $,

$$ \tag{* } | P _ {n} ^ { \prime } ( x) | \leq \frac{2 M n ^ {2} }{b - a } . $$

Inequality (*) was obtained by A.A. Markov in 1889 (see [1]). The Markov inequality is exact (best possible). Thus, for $ a = - 1 $, $ b = 1 $, considering the Chebyshev polynomials

$$ P _ {n} ( x) = \cos \{ n { \mathop{\rm arc} \cos } x \} , $$

then

$$ M = 1 ,\ \ P _ {n} ^ { \prime } ( 1) = n ^ {2} , $$

and inequality (*) becomes an equality.

For derivatives of arbitrary order $ r \leq n $, Markov's inequality implies that

$$ | P _ {n} ^ {(} r) ( x) | \leq \frac{M 2 ^ {r} }{( b - a ) ^ {r} } n ^ {2} \dots ( n - r + 1 ) ^ {2} ,\ \ a \leq x \leq b , $$

which already for $ r \geq 2 $ is not exact. An exact inequality for $ P _ {n} ^ {(} r) ( x) $ was obtained by V.A. Markov [2]:

$$ | P _ {n} ^ {(} r) ( x) | \leq \ \frac{M 2 ^ {r} n ^ {2} ( n ^ {2} - 1 ^ {2} ) \dots ( n ^ {2} -( r- 1) ^ {2} ) }{( b - a ) ^ {r} ( 2 r - 1 ) !! } ,\ a \leq x \leq b . $$

References

[1]	A.A. Markov, "Selected works" , Moscow-Leningrad (1948) (In Russian)
[2]	W.A. [V.A. Markov] Markoff, "Ueber die Funktionen, die in einem gegebenen Intervall möglichst wenig von Null abweichen" Math. Ann. , 77 (1916) pp. 213–258
[3]	I.P. Natanson, "Constructive theory of functions" , 1–2 , F. Ungar (1964–1965) (Translated from Russian)

References

[a1]	E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2]	R.J. Duffin, A.C. Schaeffer, "A refinement of an inequality of the brothers Markoff" Trans. Amer. Math. Soc. , 50 (1941) pp. 517–528
[a3]	A. Schönhage, "Approximationstheorie" , de Gruyter (1971)