Chebyshev–Markov–Stieltjes inequalities

In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes.^[1] Informally, they provide sharp bounds on a measure from above and from below in terms of its first moments.

Given m₀,...,m_2m-1 ∈ R, consider the collection C of measures μ on R such that

${\displaystyle \int x^{k}d\mu (x)=m_{k}}$

for k = 0,1,...,2m − 1 (and in particular the integral is defined and finite).

Let P₀,P₁, ...,P_m be the first m + 1 orthogonal polynomials^{[clarification needed]} with respect to μ ∈ C, and let ξ₁,...ξ_m be the zeros of P_m. It is not hard to see that the polynomials P₀,P₁, ...,P_m-1 and the numbers ξ₁,...ξ_m are the same for every μ ∈ C, and therefore are determined uniquely by m₀,...,m_2m-1.

Denote

${\displaystyle \rho _{m-1}(z)=1{\Big /}\sum _{k=0}^{m-1}|P_{k}(z)|^{2}}$.

Theorem For j = 1,2,...,m, and any μ ∈ C,

${\displaystyle \mu (-\infty ,\xi _{j}]\leq \rho _{m-1}(\xi _{1})+\cdots +\rho _{m-1}(\xi _{j})\leq \mu (-\infty ,\xi _{j+1}).}$