AI 中文总结
研究埃尔德什关于非单项式整函数\(|z| = r\)上最大模点数量的问题，证明可找到有限阶且属\(\B\)类的例子，还证明了最大模集的插值定理，能让规定点位于特定函数的最大模集中。
AI 中文摘要
1964年，埃尔德什提出问题：对于非单项式整函数，当\(r\to\infty\)时，\(|z| = r\)上最大模点的数量是否能任意大。1968年，赫尔佐格和皮拉尼安肯定回答了此问题，但未对所得函数进行定量控制。我们证明可选择一个有限阶的此类例子，且它属于埃雷缅科 - 柳比奇类\(\B\)。我们还证明了最大模集的插值定理：在几何分离条件下，规定的具有两两不同模的点可被迫位于\(\B\)类某个有限阶函数的最大模集中。
英文摘要
In 1964, Erdős asked whether, for a non-monomial entire function, the number of maximum modulus points on the circle \(|z|=r\) can become arbitrarily large as $r\to\infty$. In 1968, Herzog and Piranian answered this question affirmatively, but without quantitative control on the resulting function. We prove that such an example can be chosen to have finite order and, moreover, to belong to the Eremenko--Lyubich class \(\B\). We also prove an interpolation theorem for maximum modulus sets: prescribed points with pairwise distinct moduli can be forced to lie in the maximum modulus set of some function in \(\B\), with finite order under a geometric separation condition.
Comments27 pages, 2 figures
