In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez (Remez 1936), gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials.
Let σ be an arbitrary fixed positive number.
Define the class of polynomials πn(σ) to be those polynomials p of degree n for which on some set of measure ≥ 2 contained in the closed interval [−1, 1+σ].
Then the Remez inequality states that where Tn(x) is the Chebyshev polynomial of degree n, and the supremum norm is taken over the interval [−1, 1+σ].
, hence The R.i., combined with an estimate on Chebyshev polynomials, implies the following corollary: If J ⊂ R is a finite interval, and E ⊂ J is an arbitrary measurable set, then for any polynomial p of degree n. Inequalities similar to (⁎) have been proved for different classes of functions, and are known as Remez-type inequalities.
One important example is Nazarov's inequality for exponential sums (Nazarov 1993): In the special case when λk are pure imaginary and integer, and the subset E is itself an interval, the inequality was proved by Pál Turán and is known as Turán's lemma.
in the following way for some A > 0 independent of p, E, and n. When a similar inequality holds for p > 2.
One of the corollaries of the Remez inequality is the Pólya inequality, which was proved by George Pólya (Pólya 1928), and states that the Lebesgue measure of a sub-level set of a polynomial p of degree n is bounded in terms of the leading coefficient LC(p) as follows: