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

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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 Mazya in a recent paper found the representation for the function $H {n,p}z$ in the inequality $$|f^{n} z|\leq H {n,p}z|\Ref-\mathcal P l| {h^p\mathbf U}, \Re f\in h^p\mathbf U, z\in \mathbf U,$$ 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 Mazya.

Autor: David Kalaj; Noam D. Elkies

Fuente: https://archive.org/