Pólya inequality

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 π_n(σ) to be those polynomials p of the nth degree for which

on some set of measure ≥ 2 contained in the closed interval [−1, 1+σ]. Then the Remez inequality states that

where T_n(x) is the Chebyshev polynomial of degree n, and the supremum norm is taken over the interval [−1, 1+σ].

Observe that T_n is increasing on ${\displaystyle [1,+\infty ]}$, 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):

Let

be an exponential sum (with arbitrary λ_k ∈C), and let J ⊂ R be a finite interval, E ⊂ J — an arbitrary measurable set. Then

where C>0 is a numerical constant.

In the special case when λ_k 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.

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