Borel-carathéodory type theorem for monogenic functions (Q1024743)

Let \(f\) be analytic in \(U_R=\{z:|z|\leq R\}\). Then, for every \(r\), \(0<r<R\), the following inequality holds \[ \max_{|z|\leq r}|f(z)|\leq\frac {2r}{R-r}\sup_{|z|\leq R}|\text{Re}\,f(z)|+\frac{R+r}{R-r}|f(0)|. \] In this paper the authors generalize this known Borel-Caratheodory theorem to higher dimensions in the framework of quaterionic analysis.