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

  • A. I. Bandura Ivano-Frankivsk National Tecnical University of OIl and Gas
  • O. B. Skaskiv Ivan Franko National University of Lviv, Lviv, Ukraine
  • T. M. Salo
Keywords: entire function, bounded L-index in direction, composite function, bounded l-index

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.

Author Biographies

A. I. Bandura, Ivano-Frankivsk National Tecnical University of OIl and Gas

Ivano-Frankivsk National Tecnical University of OIl and Gas

O. B. Skaskiv, Ivan Franko National University of Lviv, Lviv, Ukraine

Ivan Franko National University of Lviv, Lviv, Ukraine

T. M. Salo
Lviv Politechnic National University, Lviv, Ukraine    

References

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Published
2021-03-04
How to Cite
1.
Bandura AI, Skaskiv OB, Salo TM. Note on composition of entire functions and bounded $L$-index in direction. Mat. Stud. [Internet]. 2021Mar.4 [cited 2021Nov.28];55(1):51-6. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/185
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