On the precise form of the inverse Markov factor for convex sets

2025/05/19 by М. А. Комаров, Komarov, Mikhail A.
Mathematics · #41A17 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2505.13285

openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K⊂ ℂ be a convex compact set, and let Π_n(K) be the class of polynomials of exact degree n, all of whose zeros lie in K. The Turán type inverse Markov factor is defined by M_n(K)=inf_{P∈ Πn(K)} (‖P'‖_C(K)/‖P‖_C(K)). A combination of two well-known results due to Levenberg and Poletsky (2002) and Révész (2006) provides the lower bound M_n(K)≥ c(wn/d²+√(n)/d), c:=0.00015, where d>0 is the diameter of K and w≥ 0 is the minimal width (the smallest distance between two parallel lines between which K lies). We prove that this bound is essentially sharp, namely, M_n(K)≤ 28(wn/d²+√(n)/d) for all n,w,d.

Citations

• The growth order of the optimal constants in Turán-Erőd type inequalities in L^q(K,μ)

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