Entire Bivariate Functions of Exponential Type II

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.