# 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)