Remark about the maximum of the modulus of an analytic function on the boundary
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 32, Tome 315 (2004), pp. 149-154
Cet article a éte moissonné depuis la source Math-Net.Ru
Let $\Lambda^{\alpha}$ be the analytic Hölder class in the unit disc $\mathbb D$. For $f\in \Lambda^{\alpha}$ and $I\subset\partial\mathbb D$, let $M_f(I)=\max_I|f|$. Assume that $I$, $J$ are arcs such that $|J|=2|I|$, $J$ and $I$ have common middle point. Then $$ M_f(J)\le C(\alpha,f)\frac{|I|^{\alpha}+M_f(I)}{\log^{\alpha}\Bigl(\frac{|I|^{\alpha}}{M_f(I)}+2\Bigr)}. $$ It is proved that this estimate cannot be improved.
@article{ZNSL_2004_315_a10,
author = {N. A. Shirokov},
title = {Remark about the maximum of the modulus of an analytic function on the boundary},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {149--154},
year = {2004},
volume = {315},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_315_a10/}
}
N. A. Shirokov. Remark about the maximum of the modulus of an analytic function on the boundary. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 32, Tome 315 (2004), pp. 149-154. http://geodesic.mathdoc.fr/item/ZNSL_2004_315_a10/