Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$ Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $\forall z \in \mathbb{U},\ Q(z) \neq 0$. For $t \in [-\pi, \pi]$, we set $f(t) = F(e^{it}) = g(t) + ih(t)$. For $u \in [-\pi, \pi]$, we define $f_{u}(t) = g(t)\cos(u) + h(t)\sin(u)$. Express the integral $$\int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}u\right) \mathrm{d}t$$ in terms of $V(f)$.
Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set
$$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $\forall z \in \mathbb{U},\ Q(z) \neq 0$. For $t \in [-\pi, \pi]$, we set $f(t) = F(e^{it}) = g(t) + ih(t)$. For $u \in [-\pi, \pi]$, we define $f_{u}(t) = g(t)\cos(u) + h(t)\sin(u)$.
Express the integral
$$\int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}u\right) \mathrm{d}t$$
in terms of $V(f)$.