We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Show that, for all $z \in \mathbb{C}$: $$|P(z)| \leq \|P\|_{\partial\mathbb{D}} \max\{1, |z|\}^d.$$ One may apply question 3.18 to the polynomials $P(X)$ and $Q(X) = X^d P\left(X^{-1}\right)$.

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity.
Show that there exist $u$ and $v$ in $\partial\mathbb{D}$ such that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq |\lambda| \cdot \prod_{i=1}^{n+m} \max\left\{\left|u - \alpha_i\right|, \left|v - \alpha_i\right|\right\}.$$

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity.
Deduce from question 3.20 that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq M(S)$$ where $S$ is the polynomial defined by: $$S(X) = (X-1)^{m+n} P\left(\frac{uX - v}{X - 1}\right)$$

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$. We set $w = \frac{v}{u}$. Show that: $$M(S) \leq \|P\|_{\mathbb{D}} \exp\left(\frac{n+m}{2\pi} \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} - w\right|\right\}\right) d\theta\right)$$

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$. We set $C = \exp\left(\frac{I}{2\pi}\right)$ with: $$I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} + 1\right|\right\}\right) d\theta$$ Using the previous questions, show that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq C^{n+m} \|P\|_{\mathbb{D}}$$

Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We are given two distinct real numbers $c$ and $d$ and we set: $$J = \begin{cases} [c,d] & \text{if } c < d \\ [d,c] & \text{if } d < c. \end{cases}$$ Let $A \in \mathbb{C}_n[X]$ and $B \in \mathbb{C}_m[X]$ be two non-zero polynomials. Show that there exist polynomials $C \in \mathbb{C}_n[X]$ and $D \in \mathbb{C}_m[X]$ satisfying the following properties: $$\begin{gathered} \|A\|_I = \|C\|_J, \quad \|B\|_I = \|D\|_J, \quad \|AB\|_I = \|CD\|_J \\ A(a) = C(c), \quad B(b) = D(d). \end{gathered}$$

Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We recall: $$C_{n,m}^I = \sup\left\{\left.\frac{\|Q\|_I \|R\|_I}{\|QR\|_I}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Deduce from question 4.27 that the quantity $C_{n,m}^I$ does not depend on the segment $I$.

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. We fix an extremal pair $(Q_0, R_0)$, i.e., $Q_0$ and $R_0$ are monic and $\frac{\|Q_0\|_I \|R_0\|_I}{\|Q_0 R_0\|_I} = C_{n,m}$.
Let $J$ be a segment contained in $I$ such that $\left\|Q_0\right\|_J = \left\|Q_0\right\|_I$ and $\left\|R_0\right\|_J = \left\|R_0\right\|_I$. Show that: $$\left\|Q_0 R_0\right\|_J = \left\|Q_0 R_0\right\|_I$$

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. We fix an extremal pair $(Q_0, R_0)$. Deduce from questions 4.27 and 4.29 that there exists an extremal pair $(Q_1, R_1)$ such that: $$\left\|Q_1\right\|_I = \left|Q_1(-1)\right| \quad \text{and} \quad \left\|R_1\right\|_I = \left|R_1(1)\right|$$

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Let $(Q_1, R_1)$ be an extremal pair such that $\left\|Q_1\right\|_I = \left|Q_1(-1)\right|$ and $\left\|R_1\right\|_I = \left|R_1(1)\right|$. Let $n_1$ and $m_1$ be the degrees of $Q_1$ and $R_1$ respectively. We set $Q_2 = X^{n-n_1} Q_1$ and $R_2 = X^{m-m_1} R_1$.
Show that $(Q_2, R_2)$ is a good extremal pair, i.e., $Q_2$ and $R_2$ are monic of degrees $n$ and $m$ respectively, $\frac{\|Q_2\|_I \|R_2\|_I}{\|Q_2 R_2\|_I} = C_{n,m}$, and $\|Q_2\|_I = |Q_2(-1)|$, $\|R_2\|_I = |R_2(1)|$.

Let $M \in \mathcal{B}_{n}$. We define $$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$ where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$.
We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$.
By using the previous question, integration by parts and inequality (3) from question $17$, show that $$\forall k \in \mathbf{N}, \quad \left|X^{T}M^{k}Y\right| \leq \frac{r^{k+1}}{(k+1)(r-1)} n b^{\prime}(M).$$

We propose to show by contradiction the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right).$$ We suppose that $\mathcal{P}$ is false. Show the existence of a sequence of natural integers $(n_{p})_{p \in \mathbb{N}}$ and a sequence of complex numbers $(z_{p})_{p \in \mathbb{N}}$ such that: $$\mathrm{e}^{z_{p}} \underset{p \rightarrow +\infty}{\rightarrow} 1 \quad \text{and} \quad \forall p \in \mathbb{N},\, |z_{p}| = (2n_{p}+1)\pi.$$

We suppose that the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$ is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$. For all $p \in \mathbb{N}$, we denote $a_{p} = \operatorname{Re}(z_{p})$ and $b_{p} = \operatorname{Im}(z_{p})$. Show that $a_{p} \underset{p \rightarrow +\infty}{\rightarrow} 0$ and $|z_{p}| - |b_{p}| \underset{p \rightarrow +\infty}{\rightarrow} 0$.

We suppose that the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right)$$ is false, and let $(n_p)_{p\in\mathbb{N}}$, $(z_p)_{p\in\mathbb{N}}$ be sequences such that $\mathrm{e}^{z_p} \to 1$ and $|z_p| = (2n_p+1)\pi$ for all $p$, with $a_p = \operatorname{Re}(z_p)$, $b_p = \operatorname{Im}(z_p)$. For all $p \in \mathbb{N}$, we denote $$\varepsilon_{p} = \begin{cases} +1 & \text{if } b_{p} \geqslant 0 \\ -1 & \text{if } b_{p} < 0 \end{cases}$$ By studying $\exp(z_{p} - \mathrm{i}\varepsilon_{p}|z_{p}|)$, reach a contradiction and conclude that $\mathcal{P}$ is true.

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, let $$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t$$ where $\gamma_{n}(t) = (2n+1)\pi\,\mathrm{e}^{2\mathrm{i}\pi t}$. Using the property $\mathcal{P}$: $$\exists c > 0,\, \forall n \in \mathbb{N},\, \forall z \in \mathbb{C},\, \left(|z| = (2n+1)\pi \Rightarrow |\mathrm{e}^{z} - 1| \geqslant c\right),$$ show that there exist two constants $a, b \in \mathbb{R}_{+}^{*}$ such that, for all $n \in \mathbb{N}^{*}$ and all $z \in \mathbb{C}$, $$|Q_{n}(z)| \leqslant a\,\mathrm{e}^{bn|z|}.$$

Let $S$ be the set of all complex numbers $z$ satisfying $\left| z ^ { 2 } + z + 1 \right| = 1$. Then which of the following statements is/are TRUE?
(A) $\left| z + \frac { 1 } { 2 } \right| \leq \frac { 1 } { 2 }$ for all $z \in S$
(B) $| z | \leq 2$ for all $z \in S$
(C) $\left| z + \frac { 1 } { 2 } \right| \geq \frac { 1 } { 2 }$ for all $z \in S$
(D) The set $S$ has exactly four elements

The greatest value of $c \in R$ for which the system of linear equations $x - cy - cz = 0$, $cx - y + cz = 0$, $cx + cy - z = 0$ has a non-trivial solution, is
(1) $-1$
(2) $2$
(3) $\frac{1}{2}$
(4) $0$