If f is an entire function and a is a complex number, a is said to be an asymptotic value of f if there exists a path \(\gamma \) from 0 to infinity such that \(f(z) - a\) tends to 0 as z tends to infinity along \(\gamma \). The Denjoy–Carleman–Ahlfors Theorem asserts that if f has n distinct asymptotic values, then the rate of growth of f is at least order n/2, mean type. A long-standing problem asks whether this conclusion holds for entire functions having n distinct asymptotic (entire) functions, each of growth at most order 1/2, minimal type. In this paper conditions on the function f and associated asymptotic paths are obtained that are sufficient to guarantee that f satisfies the conclusion of the Denjoy–Carleman–Ahlfors Theorem. In addition, for each positive integer n, an example is given of an entire function of order n having n distinct, prescribed asymptotic functions, each of order less than 1/2.

如果f是一个完整的函数并且a是一个复数，如果存在从 0 到无穷大的路径\(\gamma \)使得\(f(z) - a\)，则称a是f的渐近值趋于 0，因为z沿着\(\gamma \)趋于无穷大。Denjoy-Carleman-Ahlfors 定理断言，如果f具有n 个不同的渐近值，则f的增长率至少为n / 2阶，均值类型。一个长期存在的问题是这个结论是否适用于具有n 的整个函数不同的渐近（整个）函数，每个函数的增长最多为 1/2 阶，最小类型。在本文中，函数f 的条件和相关的渐近路径足以保证f满足 Denjoy-Carleman-Ahlfors 定理的结论。此外，对于每一个正整数Ñ，给出一个例子的顺序的整个功能的ñ具有Ñ不同，规定的渐近函数，每个阶数小于1/2。