Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties

Abstract

Let $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ with the unit ball $\mathbb{B}^n=\{z\in\mathbb{C}^: \ |z|:=\sqrt{|z|_1^2+\ldots+|z_n|^2}<1\}$ for any $z^0\in\mathbb{B}^n$. For this class of functions we consider the concept of boundedness of $L$-index in the direction $\mathbf{b},$ where $\mathbf{L}: \mathbb{B}^n\to\mathbb{R}_+$ is a positive continuous function such that $L(z)>\frac{\beta|\mathbf{b}|}{1-|z|}$ and $\beta>1$ is some constant.For functions from this class we deduce analog of Hayman's Theorem. It is criterion useful in applications todifferential equations. We introduce a concept of function having bounded value $L$-distribution in direction forthe slice holomorphic functions in the unit ball. It is proved that slice holomorphic function in the unit ball has bounded value $L$-distribution in a direction if and only if its directional derivative has bounded $L$-index in the same direction. Other propositions concern existence theorems. We show that for any slice holomorphic function $F$ with bounded multiplicities of zeros on any slice in the fixed direction there exists such a positive continuous function $L$that the function $F$ has bounded $L$-index in the direction.

Theory of Analytic Functions of Bounded Index: M. M. Sheremeta's Ideas and Its Further Development in a Multidimensional Case

“…Çàçíà÷èìî, ùî ó áàãàòîâèìiðíîìó âèïàäêó çàñòîñóâàííÿ ëîãàðèôìi÷íîãî êðèòåðiþ äî êàíîíi÷íèõ äîáóòêiâ Âå¹ðøòðàññà íàøòîâõó¹òüñÿ íà íåïîäîëàíi äîñi àíà-ëiòè÷íi òðóäíîùi, ïîâ'ÿçàíi ç îöiíêîþ ìîäóëÿ ëîãàðèôìi÷íî¨ïîõiäíî¨çà íàïðÿìîì (÷è çà êîaeíî¨çìiííîþ íàðiçíî) çîâíi äåÿêî¨âèíÿòêîâî¨ìíîaeèíè, ïîðîäaeåíî¨íóëüîâèìè òî÷êàìè ïî÷àòêîâî¨ôóíêöi¨. Òîìó íàðàçi âiäîìi óìîâè ñóòî äëÿ öiëèõ ôóíêöié ç ïëîñêèìè íóëÿìè ç îäíèì iñòîòíèì îáìåaeåííÿì óñi ãiïåðïëîùèíè íóëiâ ïàðàëåëüíi ìiae ñîáîþ [4,50].…”