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

TL;DR

This paper derives sharp bounds for higher derivatives of analytic functions in the unit disk with bounded real part, extending previous results and providing new inequalities related to Hardy spaces and harmonic functions.

Contribution

The paper finds the exact sharp constant in a key inequality for higher derivatives, extending previous work from the case n=1 to general n, and improves existing bounds.

Findings

Derived the sharp constant C_{p,n} for the inequality involving higher derivatives.

Extended previous results from first derivatives to higher derivatives.

Provided new inequalities for the modulus of higher derivatives of analytic functions.

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 $$|f^{(n)} (z)|\leq H_{n,p}(z)|\Re(f-\mathcal P_l)|_{h^p(\mathbf U)}, \Re f\in h^p(\mathbf U), z\in \mathbf U,$$ 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 Markovi\'c, 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.