Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function: $$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$ where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$ and $M(Q) = \exp\left(\frac{1}{2\pi}\int_0^{2\pi}\ln\left|Q(e^{i\theta})\right|d\theta\right)$. Compute the limit of $\varphi'$ at $0^+$ then deduce that: $$M_p(Q)^{1/p} \underset{p \rightarrow 0^+}{\longrightarrow} M(Q).$$
Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function:
$$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$
where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$ and $M(Q) = \exp\left(\frac{1}{2\pi}\int_0^{2\pi}\ln\left|Q(e^{i\theta})\right|d\theta\right)$. Compute the limit of $\varphi'$ at $0^+$ then deduce that:
$$M_p(Q)^{1/p} \underset{p \rightarrow 0^+}{\longrightarrow} M(Q).$$