For any $\alpha \in \mathbb {R},$ we consider the weighted Taylor shift operators $T_{\alpha }$ acting on the space of analytic functions in the unit disc given by $T_{\alpha }:H(\mathbb {D})\rightarrow H(\mathbb {D}),$ $$ \begin{align*}f(z)=\sum_{k\geq 0}a_{k}z^{k}\mapsto T_{\alpha}(f)(z)=a_1+\sum_{k\geq 1}\Big(1+\frac{1}{k}\Big)^{\alpha}a_{k+1}z^{k}.\end{align*}$$ We establish the optimal growth of frequently hypercyclic functions for $T_\alpha $ in terms of $L^p$ averages, $1\leq p\leq +\infty $ . This allows us to highlight a critical exponent.

对于任何 $\alpha \in \mathbb {R},$ 我们考虑加权泰勒移位算子 $T_{\alpha }$ 作用于 $T_{\alpha }:H( \mathbb {D})\rightarrow H(\mathbb {D}),$ $$ \begin{align*}f(z)=\sum_{k\geq 0}a_{k}z^{k}\mapsto T_{\alpha}(f)(z)=a_1+\sum_{k\geq 1}\Big(1+\frac{1}{k}\Big)^{\alpha}a_{k+1}z^ {k}.\end{align*}$$ 我们根据 $L^p$ 平均值 $1\leq p\leq +\infty $ 为 $T_\alpha $ 建立频繁超循环函数的最优增长。这使我们能够突出显示关键指数。