Geodesic
Smoothness of a holomorphic function and its modulus on the boundary of a polydisk
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 172-176 
N. A. Shirokov. Smoothness of a holomorphic function and its modulus on the boundary of a polydisk. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 45, Tome 456 (2017), pp. 172-176. http://geodesic.mathdoc.fr/item/ZNSL_2017_456_a14/
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     author = {N. A. Shirokov},
     title = {Smoothness of a~holomorphic function and its modulus on the boundary of a~polydisk},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_456_a14/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We prove that if a function $f$ is holomorphic in the polydisk $\mathbb D^n$, $n\ge2$, $f$ is continuous in $\overline{\mathbb D^n}$, $f(z)\ne0$, $z\in\mathbb D^n$, and $|f|$ belongs to the $\alpha$-Hölder class, $0<\alpha<1$, on the boundary of $\mathbb D^n$ then $f$ belongs to the $(\frac\alpha2-\varepsilon)$-Hölder class on $\overline{\mathbb D^n}$ for any $\varepsilon>0$.

[1] N. A. Shirokov, “Gladkost golomofornoi v share funktsii i ee modulya na sfere”, Zap. nauchn. semin. POMI, 447, 2016, 123–128

[2] N. A. Shirokov, “Klassy Gëldera v oblastyakh Lavrenteva”, Zap. nauchn. semin. POMI, 282, 2001, 256–275 | MR | Zbl

[3] U. Rudin, Teoriya funktsii v edinichnom share v $\mathbb C^n$, Mir, Moskva, 1984 | MR