We obtain the optimal boundary behavior of the log-frequently hypercyclic functions with respect to the Taylor shift acting on $H(\mathbb{D})$ in terms of average $L^p$-norms. In passing we establish some new results on the growth of frequently or log-frequently hypercyclic functions for the differentiation operator on $H(\mathbb{C})$. All these results highlight the similarities and the differences between the lower and upper bounds on the growth of frequently and log-frequently hypercyclic functions, on the one hand in the case of the Taylor shift operator on $H(\mathbb{D})$ and on the other hand in the case of the differentiation operator on $H(\mathbb{C})$.

中文翻译：

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频繁或对数频繁的超循环函数的增长

我们获得了对数频繁超循环函数的最优边界行为，关于泰勒移位作用于 $H(\mathbb{D})$ 就平均而言 ${升}^{磷}$-规范。顺便说一下，我们建立了一些关于微分算子的频繁或对数频繁超循环函数增长的新结果$H(\mathbb{C})$. 所有这些结果都突出了频繁和对数频繁超循环函数增长的下界和上限之间的异同，一方面在泰勒移位算子的情况下$H(\mathbb{D})$ 另一方面，在微分算子的情况下 $H(\mathbb{C})$.