Inégalités de type Remez dans le plan

We obtain a sharp upper bound on the modulus of a complex polynomial ppzq at any boundary point of a domain G € C bounded, the area of the subset of G, where |ppzq| ¤ 1, is known. Remez’s inequality makes it possible to give an increase on the supremum norms of certain polynomials, the increase being reached by the Chebyshev polynomials. Similar inequalities have been proven for different classes of functions. an important example is the Nazarov inequality for exponential sums. One of the corollaries of the Remez inequality is the Pòlya inequality which states that the Lebesgue measure of a set of sublevels of a polynomial of degree n is bounded according to its dominant coefficient. Keywords : Remez inequalities, Polynomial, Green’s function.