Note on boundedness of the $L$-index in the direction of the composition of slice entire functions

Abstract

We study a composition of two functions belonging to a class of slice holomorphic functions in the whole $n$-dimensional complex space. The slice holomorphy in the space means that for some fixed direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ and for every point $z^0\in\mathbb{C}^n$ the function is holomorphic on its restriction on the slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}.$ An additional assumption on joint continuity for these functions allows to construct an analog of theory of entire functions having bounded index. The analog is applicable to study properties of slice holomorphic solutions of directional differential equations, describe local behavior and value distribution.In particular, we found conditions providing boundedness of $L$-index in the direction $\mathbf{b}$ for a function $f(\underbrace{\Phi(z),\ldots,\Phi(z)}_{m\text{ times}}),$where $f: \mathbb{C}^n\to\mathbb{C}$ is a slice entire function, $\Phi: \mathbb{C}^n\to\mathbb{C}$ is a slice entire function,${L}: \mathbb{C}^n\to\mathbb{R}_+$ is a continuous function.The obtained results are also new in one-dimensional case, i.e. for $n=1,$ $m=1.$ They are deduced using new approach in this area analog of logarithmic criterion.For a class of nonvanishing outer functions in the composition the sufficient conditions obtained by logarithmic criterion are weaker than the conditions by the Hayman theorem.

L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient

“…Many papers are devoted to a composition of two holomorphic functions belonging to different classes and various definitions of the index. Nowadays, the most exhaustively investigated cases are case of the bounded l-index for entire functions of single variable [8,10,11] and analytic in a disc functions [12,13] and the case of the bounded L-index in a direction for multivariate entire functions [14] and analytic functions in a unit ball [15].…”

“…But the authors studied composition in the case that every firstorder partial derivative of the inner function does not equal zero. Recently, for the finite directional L-index, an approach was developed [14] to study the composition without the condition that the zero set of the first-order directional derivative of the inner function is empty.…”

On an attempt to introduce a notion of bounded index for the Fueter regular functions of the quaternionic variable

“…The notion was firstly appeared for entire functions of single complex variable [10]. These functions have applications in the analytic theory of differential equations [16] and value distribution theory [2,6,17]. Moreover, there are many papers on different multidimensional complex analogs of the notion [3][4][5]13].…”

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

“…Íàñàìêiíåöü çàçíà÷èìî, ùî Ì. Ì. Øåðåìåòà [54,96,106] òàêîae ó ðàìêàõ òåîðiô óíêöié îáìåaeåíîãî iíäåêñó ïåðøèì ïî÷àâ âèâ÷àòè óìîâè, çà ÿêèõ êîìïîçèöiÿ àíà-ëiòè÷íèõ ôóíêöié ìàòèìå îáìåaeåíèé iíäåêñ. Öÿ åñòàôåòà äîñëiäaeåííÿ êîìïîçèöié ó áàãàòîâèìiðíîìó âèïàäêó âiä íüîãî ïiäõîïëåíà ó ñòàòòÿõ [28,30,43].…”