PROPERTIES OF CLASSES OF SLICE ENTIRE FUNCTIONS AND SLICE HOLOMORPHIC IN THE UNIT BALL FUNCTIONS
DOI:
https://doi.org/10.31471/2304-7399-2021-16(60)-7-10Keywords:
slice entire function, slice holomorphic function, several complex variables, unit ball, directional derivativeAbstract
In the paper we investigate properties of class of slice entire functions of several complex variables i.e. these functions are entire on every slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ for an arbitrary
$z^0\in\mathbb{C}^n$ and for the fixed direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$.
For a function $F$ from this class we consider a slice function $g_z(t)=F(z+t\mathbf{b})$ $(z\in\mathbb{C}^n,$ $t\in\mathbb{C})$ and a directional derivative
$\partial_{\mathbf{b}}F(z):=g'_z(0),$ $\partial^p_{\mathbf{b}}F(z):=\partial_{\mathbf{b}}(\partial_{\mathbf{b}}^{p-1}F(z)),$ $p\ge 2.$
We show that if a function $F$ belongs to this class then for any $p\in\mathbb{N}$ the function $\partial_{\mathbf{b}}^p F$ also belongs to the class.
A similar results is also obtained for functions which are slice holomorphic in the unit ball.
References
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Bandura, A.; Martsinkiv, M.; Skaskiv, O. Slice Holomorphic Functions in the Unit Ball Having a Bounded L-Index in Direction. Axioms 2021, 10 (1), 4. https://doi.org/10.3390/axioms10010004
