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

Abstract

Let $n$ be a positive integer. Let $\mathbf{U}$ be the unit disk, $p\ge 1$ and let $h^p(\mathbf{U})$ be the Hardy space of harmonic functions. Kresin and Maz'ya in a recent paper found the representation for the function $H_{n,p}(z)$ in the inequality $$|f^{(n)}(z)|\le H_{n,p}(z)|\Re (f-{\mathcal{P}}_l){|}_{h^p(\mathbf{U})},\Re f\in h^p(\mathbf{U}),z\in \mathbf{U},$$ where ${\mathcal{P}}_l$ is a polynomial of degree $l\le n-1$. We find or represent the sharp constant $C_{p,n}$ in the inequality $H_{n,p}(z)\le \frac{C_{p,n}}{(1-|z{|}^2{)}^{1/p+n}}$. This extends a recent result of the second author and Marković, where it was considered the case $n=1$ only. As a corollary, an inequality for the modulus of the $n-th$ derivative of an analytic function defined in a complex domain with the bounded real part is obtained. This result improves some recent result of Kresin and Maz'ya.

References in corpus (1)

• Optimal estimates for the gradient of harmonic functions in the unit disk