2016/11/15 by Polina Yu. Glazyrina, Glazyrina, Polina Yu., Szilárd Gy. Révész +1
Mathematics · #30E10 #41A17 (Primary) #52A10 (Secondary) #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.CV #msc:30E10 #msc:41A17 #msc:52A10
paper · pdf · doi:10.48550/arxiv.1611.04897
arXiv admin note: text overlap with arXiv:1512.08268
arxiv created 2016/11/15 · openalex publication_date 2016/11/15 · arxiv updated 2016/11/16 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
P. Turán was the first to derive lower estimations on the uniform norm of the derivatives of polynomials p of uniform norm 1 on the interval I:=[-1,1] and the disk D:=\z ∈ C~:~|z| ≤ 1\, under the normalization condition that the zeroes of the polynomial p in question all lie in I or D, resp. Namely, in 1939 he proved that with n:=deg p tending to infinity, the precise growth order of the minimal possible derivative norm is √(n) for I and n for D. Already the same year J. Erőd considered the problem on other domains. In his most general formulation, he extended Turán's order n result on D to a certain general class of piecewise smooth convex domains. Finally, a decade ago the growth order of the minimal possible norm of the derivative was proved to be n for all compact convex domains. Turán himself gave comments about the above oscillation question in Lq norm on D. Nevertheless, till recently results were known only for I, D and so-called R-circular domains. Continuing our recent work, also here we investigate the Turán-Erőd problem on general classes of domains.