2024/07/25 by Polina Yu. Glazyrina, Glazyrina, Polina, Szilárd Gy. Révész +1
Computer Science · Mathematics · #41A17 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2407.18404
openalex publication_date 2024/07/25 · openalex created_date 2024/09/12 · openalex updated_date 2026/07/28
In 1939 Pál Turán and János Erőd initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a matter of normalization they considered the family Pn(K) of degree n polynomials with all zeros lying in the given convex, compact subset K\Subset \mathbb C. While Turán obtained the first results for the interval I:=[-1,1] and the disk D:=\ z∈ \mathbb C~:~ |z|≤ 1\, Erőd extended investigations to other compact convex domains, too. The order of the optimal constant was found to be √(n) for I and n for D. It took until 2006 to clarify that all compact convex domains (with nonempty interior), follow the pattern of the disk, and admit an order n inequality. For Lq(∂ K) norms with any 1≤ q <∞ we obtained order n results for various classes of domains. Further, in the generality of all convex, compact domains we could show a c n/log n lower bound together with an O(n) upper bound for the optimal constant. Also, we conjectured that all compact convex domains admit an order n Turán type inequality. Here we prove this for all polygonal convex domains and any 0< q <∞.