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.