For a transcendental entire function f, the property that there exists \(r>0\) such that \(m^n(r)\rightarrow \infty \) as \(n\rightarrow \infty \), where \(m(r)=\min \{|f(z)|:|z|=r\}\), is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r).

对于超越整个函数 f，存在\(r>0\)使得\(m^n(r)\rightarrow \infty \)为\(n\rightarrow \infty \) 的性质，其中\(m (r)=\min \{|f(z)|:|z|=r\}\)，与 Eremenko 和 Baker 的猜想有关，对于这两个 1/2 阶极小类型是一个显着的比率生长。我们表明，如果f的最大模数具有足够的规律增长，则该性质适用于 1/2 阶极小类型的函数， 并且我们通过使用 Kjellberg 构造小函数的整个函数的最近概括来举例说明我们的结果的尖锐性。增长，这允许相当精确地控制m ( r）。