2019/09/23 by Tamás Erdélyi, Erdélyi, Tamás
Mathematics · #41A17 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions #math.CA #msc:41A17
paper · pdf · doi:10.48550/arxiv.1909.10118
arxiv created 2019/09/23 · openalex publication_date 2019/09/23 · arxiv updated 2019/09/24 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
Let \cal Pnc denote the set of all algebraic polynomials of degree at most n with complex coefficients. Let D+ := \z ∈ ℂ: |z| ≤ 1, \Im(z) ≥ 0\ be the closed upper half-disk of the complex plane. For integers 0 ≤ k ≤ n let \mathcal Fn,kc be the set of all polynomials P ∈ \mathcal Pnc having at least n-k zeros in D+. Let ‖f‖A := supz ∈ A|f(z)| for complex-valued functions defined on A ⊂ \Bbb C. We prove that there are absolute constants c1 > 0 and c2 > 0 such that c1 ((n)/(k+1))1/2 ≤ infP\frac‖P′‖[-1,1]‖P‖[-1,1] ≤ c2 ((n)/(k+1))1/2 for all integers 0 ≤ k ≤ n, where the infimum is taken for all 0 \not≡ P ∈ \mathcal Fn,kc having at least one zero in [-1,1]. This is an essentially sharp reverse Markov-type inequality for the classes \mathcal Fn,kc extending earlier results of Turán and Komarov from the case k=0 to the cases 0 ≤ k ≤ n.