Note on composition of entire functions and bounded $L$-index in direction

Abstract

We study the following question: ``Let $f\colon \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi\colon \mathbb{C}^n\to \mathbb{C}$ an be entire function, $n\geq2,$ $l\colon \mathbb{C}\to \mathbb{R}_+$ be a continuous function. What is a positive continuous function $L\colon \mathbb{C}^n\to \mathbb{R}_+$ and a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ such that the composite function $f(\Phi(z))$ has bounded $L$-index in the direction~$\mathbf{b}$?'' In the present paper, early known result on boundedness of $L$-index in direction for the composition of entire functions $f(\Phi(z))$ is modified. We replace a condition that a directional derivative of the inner function $\Phi$ in a direction $\mathbf{b}$ does not equal zero. The condition is replaced by a construction of greater function $L(z)$ for which $f(\Phi(z))$ has bounded $L$-index in a direction. We relax the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K|\partial_{\mathbf{b}}\Phi(z)|^k$ for all $z\in\mathbb{C}^n$,where $K\geq 1$ is a constant and ${\partial_{\mathbf{b}} F(z)}:=\sum\limits_{j=1}^{n}\!\frac{\partial F(z)}{\partial z_{j}}{b_{j}}, $ $\partial_{\mathbf{b}}^k F(z):=\partial_{\mathbf{b}}\big(\partial_{\mathbf{b}}^{k-1} F(z)\big).$ It is replaced by the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K(l(\Phi(z)))^{1/(N(f,l)+1)}|\partial_{\mathbf{b}}\Phi(z)|^k,$ where $N(f,l)$ is the $l$-index of the function $f.$The described result is an improvement of previous one.

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

“…Among many papers on composition of entire functions and index boundedness [5,9,10,19,28] we should like to mention paper [4] because it is also devoted slice entire functions. There was investigated some composition of slice entire functions by usage the analog of Hayman's Theorem for this class of function, i.e.…”

“…Notations and definitions. Let us introduce some notations from [1] (see also [10,11]). Let R + = (0, +∞), R * + = [0, +∞), 0 = (0, .…”

“…2. Introduction and formulation of the problem Despite many studies on composition of entire functions and slice entire functions and boundedness of L-index in direction and boundedness of L-index in joint variables [4,5,9,10,19,28], this topic is still interested because it allows to discover new properties of functions having bounded index and apply it to differential equations and value distribution theory. In particular, if an entire function has a bounded index in some sense, then multiplicities of its zeros are uniformly bounded.…”

“…However, in some cases all known results on composition (for example, see below Theorems 1 and 2) are not applicable even if we consider two entire functions of one complex variable with bounded multiplicities of zeros. Among many papers on composition of entire functions and index boundedness [5,9,10,19,28] we should like to mention paper [4] because it is also devoted slice entire functions. There was investigated some composition of slice entire functions by usage the analog of Hayman's Theorem for this class of function (see below Theorem 5).…”

“…They helped to find sufficient conditions by the coefficients of equations providing index boundedness of every analytic solution. For the composition we do not know results obtained by the logarithmic criterion (see [5,9,10,19,28]). In other words, the present paper is the first paper on application of logarithmic criterion to composition of holomorphic functions in theory of bounded index.…”