A Real Part Theorem for the Higher Derivatives of Analytic Functions in the Unit Disk

摘要

Let n be a positive integer. Let U be the unit disk, p ≥ 1 and let h~p (U) be the Hardy space of harmonic functions. Kresin and Maz'ya in a recent paper found a representation for the function H_(n,p)(z) in the inequality ∣f~(n)(z)∣ ≤ H_(n,p)(z)‖R(f - P_l)‖h~p(U), Rf ∈ h~p (U), z ∈ U, where P_l is a polynomial of degree l ≤ n - 1. We determine the sharp constant C_(p,n), in the inequality H_(n,p)(z) ≤ C_(p,n)/(1-∣z∣~2)~(1/p+n), This extends a recent result of Kalaj and Markovic, where only the case n = 1 was considered. As a corollary, an inequality for the modulus of n-th derivative of an analytic function defined in a complex domain with the bounded real part is obtained. This result improves a recent result of Kresin and Maz'ya.