Geodesic
Smoothness of a holomorphic function in a ball and smoothness of its modulus on the sphere
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 123-128 
N. A. Shirokov. Smoothness of a holomorphic function in a ball and smoothness of its modulus on the sphere. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 44, Tome 447 (2016), pp. 123-128. http://geodesic.mathdoc.fr/item/ZNSL_2016_447_a8/
@article{ZNSL_2016_447_a8,
     author = {N. A. Shirokov},
     title = {Smoothness of a holomorphic function in a ball and smoothness of its modulus on the sphere},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {123--128},
     year = {2016},
     volume = {447},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_447_a8/}
}
TY  - JOUR
AU  - N. A. Shirokov
TI  - Smoothness of a holomorphic function in a ball and smoothness of its modulus on the sphere
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2016
SP  - 123
EP  - 128
VL  - 447
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2016_447_a8/
LA  - ru
ID  - ZNSL_2016_447_a8
ER  - 
%0 Journal Article
%A N. A. Shirokov
%T Smoothness of a holomorphic function in a ball and smoothness of its modulus on the sphere
%J Zapiski Nauchnykh Seminarov POMI
%D 2016
%P 123-128
%V 447
%U http://geodesic.mathdoc.fr/item/ZNSL_2016_447_a8/
%G ru
%F ZNSL_2016_447_a8

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let a function $f$ be holomorphic in the unit ball $\mathbb B^n$, continuous in the closed ball $\overline{\mathbb B}^n$, and let $f(z)\ne0$, $z\in\mathbb B^n$. Assume that $|f|$ belongs to the $\alpha$-Hölder class on the unit sphere $S^n$, $0<\alpha\leq1$. The present paper is devoted to the proof of statement that $f$ belongs to the $\alpha/2$-Hölder class on $\overline{\mathbb B}^n$.

[1] V. P. Khavin, F. A. Shamoyan, “Analiticheskie funktsii s lipshitsevym modulem granichnykh znachenii”, Zap. nauchn. semin. LOMI, 19, 1970, 237–239 | MR | Zbl

[2] S. V. Kislyakov, lichnoe soobschenie

[3] N. A. Shirokov, Analytic functions smooth up to the boundary, Lecture Notes in Math., 1312, 1988 | DOI | MR | Zbl

[4] U. Rudin, Teoriya funktsii v edinichnom share iz $C^n$, Mir, M., 1984 | MR

[5] A. Zigmund, Trigonometricheskie ryady, v. 1, Mir, M., 1965