On real part theorem for the higher derivatives of analytic functions in\n the unit disk

2012/02/12 by David Kalaj, Noam D. Elkies, Kalaj, David +1
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1202.2520

openalex publication_date 2012/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let n be a positive integer. Let mathbf U be the unit disk, p\≥ 1 and\nlet h^p( mathbf U) be the Hardy space of harmonic functions. Kresin and\nMaz'ya in a recent paper found the representation for the function H_n,p(z)\nin the inequality |f⁽ⁿ⁾ (z)|
leq H_n,p(z)|
Re(f-
mathcal\nP_l)|_{h^p(
mathbf U)},
Re f
in h^p(
mathbf U), z
in
mathbf U, where\n mathcal P_l is a polynomial of degree l\≤ n-1. We find or represent the\nsharp constant C_p,n in the inequality H_n,p(z)\≤\n fracC_p,n(1-|z|²)^1/p+n. This extends a recent result of the second\nauthor and Markovi 'c, where it was considered the case n=1 only. As a\ncorollary, an inequality for the modulus of the n-th derivative of an\nanalytic function defined in a complex domain with the bounded real part is\nobtained. This result improves some recent result of Kresin and Maz'ya.\n

Related

• Optimal estimates for the gradient of harmonic functions in the unit disk
• Gabriel's problem for harmonic Hardy spaces
• On some Riesz and Carleman type inequalities for harmonic functions on the unit disk
• Estimates of partial derivatives for harmonic functions on the unit disc
• Explicit real-part estimates for high order derivatives of analytic\n functions