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

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

2012-02-12

arxiv.org/abs/1202.2520v2

Mathematics

Complex Variables

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

On real part theorem for the higher derivatives of analytic functions in the unit disk does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On real part theorem for the higher derivatives of analytic functions in the unit disk, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On real part theorem for the higher derivatives of analytic functions in the unit disk will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-371565