# Entire Bivariate Functions of Exponential Type II

• A. Bandura Ivano-Frankivsk National Technical University of Oil and Gas Ivano-Frankivsk, Ukraine
• F. Nuray Department of Mathematics, Afyon Kocatepe University, Afyonkarahisar, Turkey
Keywords: Entire function, bivariate function, exponential type, bounded index

### Abstract

Let $f(z_{1},z_{2})$ be a bivariate entire function and $C$ be a positive constant. If $f(z_{1},z_{2})$ satisfies the following inequality for non-negative integer $M$, for all non-negative integers $k,$ $l$ such that $k+l\in\{0, 1, 2, \ldots, M\}$, for some integer $p\ge 1$ and for all $(z_{1},z_{2})=(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})$ with $r_1$ and $r_2$ sufficiently large:
\begin{gather*}\sum_{i+j=0}^{M}\frac{\left(\int_{0}^{2\pi}\int_{0}^{2\pi}|f^{(i+k,j+l)}(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})|
^{p}d\theta_{1}d\theta_{2}\right)^{\frac{1}{p}}}{i!j!}\ge \\
\ge \sum_{i+j=M+1}^{\infty}\frac{\left(\int_{0}^{2\pi}\int_{0}^{2\pi}
|f^{(i+k,j+l)}(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})|^{p}d\theta_{1}d\theta_{2}\right)^{\frac{1}{p}}}{i!j!},
\end{gather*}
then $f(z_{1},z_{2})$ is of exponential type not exceeding
$2+2\log\Big(1+\frac{1}{C}\Big)+\log[(2M)!/M!].$
If this condition is replaced by related conditions, then also $f$ is of exponential type.

### Author Biographies

A. Bandura, Ivano-Frankivsk National Technical University of Oil and Gas Ivano-Frankivsk, Ukraine

Ivano-Frankivsk National Technical University of Oil and Gas
Ivano-Frankivsk, Ukraine

F. Nuray, Department of Mathematics, Afyon Kocatepe University, Afyonkarahisar, Turkey

Department of Mathematics, Afyon Kocatepe University,
Afyonkarahisar, Turkey

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Published
2023-06-23
How to Cite
Bandura, A., & Nuray, F. (2023). Entire Bivariate Functions of Exponential Type II. Matematychni Studii, 59(2), 156-167. https://doi.org/10.30970/ms.59.2.156-167
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