Entire functions of several complex variables bounded outside a set of finite volume (Q1096049)

Let f(z) be a transcendental entire function and \(M(r)= \max | f(z)|\) for \(| z| =r\). For simplicity, denote \(L(r)=(\log \log \log M(r))/(\log r).\) \textit{A. Edrei} and \textit{P. Erdős} [Acta Math. Hung. 45, 367-376 (1985; Zbl 0578.30018)] showed that if \(\limsup\) or the weaker assumption \(\liminf\) of L(r) is less than 2 as \(r\to +\infty\), then the area, that is, its 2-dimensional Lebesgue measure \(m(| f(z)| >B)=+\infty\) for every positive constant B. They also constructed an entire function and showed that the result no longer holds if the \(\limsup\) and \(\inf <2\) are replaced by \(\leq 2\). Here, in this paper, the authors generalize a part of the above results for the one dimensional case to the \(n\geq 2\) dimensional case. That is, for entire function f(z) on \({\mathbb{C}}^ n\), if \(\liminf_{r\to +\infty} L(r)<2n\) with \(| z| =(| z_ 1|^ 2+...+| z_ n|^ 2)^{1/2}\), then \(m_{2n}(| f(z)| >B)=+\infty\), where \(m_{2n}\) denotes the 2n dimensional Lebesgue measure.