On real part theorem for the higher derivatives of analytic functions in the unit diskReport as inadecuate



 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.



Author: David Kalaj; Noam D. Elkies

Source: https://archive.org/







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