Show that there exists a constant $C > 0$ for which the interpolation inequality $$\forall f \in \mathcal{C}^K([0,1]), \quad \max_{0 \leqslant k \leqslant K-1} \left\|f^{(k)}\right\|_\infty \leqslant \left\|f^{(K)}\right\|_\infty + C \sum_{\ell=1}^K \left|f\left(x_\ell\right)\right|$$ is satisfied.

Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$.
Let $M \in \mathcal{B}_{n}$, $r \in ]1, +\infty[$ and $(X,Y) \in \mathcal{M}_{n,1}(\mathbf{C})^{2}$. Determine a sequence of complex numbers $(c_{j})_{j \in \mathbf{N}}$ such that the series $\sum c_{j}$ converges absolutely and that $$\forall t \in \mathbf{R}, \quad X^{T}R_{re^{it}}(M)Y = \sum_{j=0}^{+\infty} c_{j} e^{-i(j+1)t}.$$ If $k \in \mathbf{N}$, deduce, using question 9, an integral expression for $X^{T}M^{k}Y$.

We fix a real $\alpha > 0$ and an integer $n \geq 1$. Subject to existence, we set
$$S _ { n , \alpha } ( t ) : = \sum _ { k = 1 } ^ { + \infty } \frac { k ^ { n } e ^ { - k t \alpha } } { \left( 1 - e ^ { - k t } \right) ^ { n } }$$
We also introduce the function
$$\varphi _ { n , \alpha } : x \in \mathbf { R } _ { + } ^ { * } \mapsto \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } }$$
Show, for all real $t > 0$, the existence of $S _ { n , \alpha } ( t )$, its strict positivity, and the identity
$$\int _ { 0 } ^ { + \infty } \varphi _ { n , \alpha } ( x ) \mathrm { d } x = t ^ { n + 1 } S _ { n , \alpha } ( t ) - \sum _ { k = 0 } ^ { + \infty } \int _ { k t } ^ { ( k + 1 ) t } ( x - k t ) \varphi _ { n , \alpha } ^ { \prime } ( x ) \mathrm { d } x$$
Deduce that
$$S _ { n , \alpha } ( t ) = \frac { 1 } { t ^ { n + 1 } } \int _ { 0 } ^ { + \infty } \frac { x ^ { n } e ^ { - \alpha x } } { \left( 1 - e ^ { - x } \right) ^ { n } } \mathrm {~d} x + O \left( \frac { 1 } { t ^ { n } } \right) \quad \text { as } t \rightarrow 0 ^ { + }$$

Let $K \in \mathbb{N}^{\star}$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
In the special case $[a,b] = [0,1]$, justify that the series $\sum f_n^{(k)}$ converges normally on $[a,b]$ for all $k \in \llbracket 0, K-1 \rrbracket$.

Let $K \in \mathbb{N}^\star$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
In the particular case $[a,b] = [0,1]$, justify that the series $\sum f_n^{(k)}$ converges normally on $[a,b]$ for all $k \in \llbracket 0, K-1 \rrbracket$.

We consider a power series $g \in O_1$ with non-negative real coefficients. We assume that there exist $a > 0, b > 0$ such that $$g \prec a\left(I + \frac{g^2}{b - g}\right)$$ and $h$ is the function from question (11). Show, by induction on $k$, that $(g)_k \leqslant (h)_k$ for all $k \in \mathbb{N}$, conclude.

Let $K \in \mathbb{N}^{\star}$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
Treat the question of showing that $\sum f_n^{(k)}$ converges normally on $[a,b]$ for all $k \in \llbracket 0, K-1 \rrbracket$ in the general case of a segment $[a,b]$ with $a < b$. One may examine $f_n \circ \sigma$ where $\sigma : [0,1] \rightarrow [a,b]$ is defined by $\sigma(t) = (1-t)a + tb$ for all $t \in [0,1]$.

Let $K \in \mathbb{N}^\star$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
Treat the previous question in the general case of a segment $[a,b]$ with $a < b$. One may examine $f_n \circ \sigma$ where $\sigma : [0,1] \rightarrow [a,b]$ is defined by $\sigma(t) = (1-t)a + tb$ for all $t \in [0,1]$.

We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$. Show that there exists a unique series $h \in O_1$ such that $h \circ f = I$, and that $(h)_1 = 1/\lambda$.

Let $K \in \mathbb{N}^{\star}$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ satisfying hypotheses (H1) and (H2). According to the result of the previous question, we set $F_k(x) = \sum_{n=0}^{+\infty} f_n^{(k)}(x)$ for all $x \in [a,b]$. Prove that $F_0$ is of class $\mathcal{C}^K$ on $[a,b]$ and that $F_0^{(k)} = F_k$ for all $k \in \llbracket 1, K \rrbracket$.

Let $K \in \mathbb{N}^\star$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
According to the result of the previous question, we can set $F_k(x) = \sum_{n=0}^{+\infty} f_n^{(k)}(x)$ for all $x \in [a,b]$. Prove that $F_0$ is of class $\mathcal{C}^K$ on $[a,b]$ and that $F_0^{(k)} = F_k$ for all $k \in \llbracket 1, K \rrbracket$.

Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$, and for all $k \in \llbracket 0 , n \rrbracket$, $z _ { k } = \cos \left( \frac { k \pi } { n } \right)$. In this question, we prove $$\sup _ { x \in [ - 1,1 ] } | W ( x ) | \geqslant \frac { 1 } { 2 ^ { n - 1 } }$$ by contradiction.

• If we assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | < \frac { 1 } { 2 ^ { n - 1 } }$, show that, for all $k \in \llbracket 0 , n - 1 \rrbracket , Q \left( z _ { k } \right) Q \left( z _ { k + 1 } \right) < 0$.
• Deduce a contradiction and conclude.

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 $f \in \mathcal{C}^{1}$ with real values. We assume that the set $C(f)$ of points in $]-\pi, \pi[$ where the function $f^{\prime}$ vanishes is finite. We denote by $\ell$ the cardinality of $C(f)$ and, if $\ell \geq 1$, we denote by $t_{1} < \cdots < t_{\ell}$ the elements of $C(f)$. We set $t_{0} = -\pi$ and $t_{\ell+1} = \pi$. For $0 \leq j \leq \ell$, let $\psi_{j}$ be the function from $\mathbf{R}$ to $\{0,1\}$ equal to 1 on $\left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$ and to 0 on $\mathbf{R} \backslash \left[f\left(t_{j}\right), f\left(t_{j+1}\right)\right[$.
If $y \in \mathbf{R}$, show that the set $f^{-1}(\{y\}) \cap [-\pi, \pi[$ is finite with cardinality bounded by $\ell+1$; we denote this cardinality by $N(y)$. If $y \in \mathbf{R}$, express $N(y)$ in terms of $\psi_{0}(y), \ldots, \psi_{\ell}(y)$. Deduce the inequality $$V(f) \leq 2\max\{N(y); y \in \mathbf{R}\}\|f\|_{\infty}$$

We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$. Show that there exists a unique series $g \in O_1$ such that $f \circ g = I$.

Explicitly determine $F^{\prime\prime}(x)$, where $F(x) = \sum_{n=1}^{+\infty} f_n(x)$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$.

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Let $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$. Explicitly state $F^{\prime\prime}(x)$.

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality.
Let $n \in \mathbf { N }$. Show that $P _ { n , N }$ is included in $\llbracket 0 , n \rrbracket ^ { N }$ and non-empty for all $N \in \mathbf { N } ^ { * }$, that the sequence $\left( p _ { n , N } \right) _ { N \geq 1 }$ is increasing and that it is constant from rank $\max ( n , 1 )$ onwards.

For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality.
Let $n \in \mathbf{N}$. Show that $P_{n,N}$ is included in $[0,n]^N$ and non-empty for all $N \in \mathbf{N}^*$, that the sequence $(p_{n,N})_{N \geq 1}$ is increasing and that it is constant from rank $\max(n,1)$ onwards.

We consider the special case of a series of the form $f = I + F, F \in O_2$. We denote $G := f^\dagger - I, G \in O_2$. Show that $[G]_{d+1} + F \circ (I + [G]_d) \in O_{d+2}$ for all $d \geqslant 1$ (the notation $[f]_d$ is defined in the introduction to the subject).

Show that $|F(x)| \leqslant \frac{1}{3}$ for all $x \in [1,2]$, where $F(x) = \sum_{n=1}^{+\infty} f_n(x)$.

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Let $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$. Show that $|F(x)| \leqslant \frac{1}{3}$ for all $x \in [1,2]$.

For $( n , N ) \in \mathbf { N } \times \mathbf { N } ^ { * }$, we denote by $P _ { n , N }$ the set of lists $\left( a _ { 1 } , \ldots , a _ { N } \right) \in \mathbf { N } ^ { N }$ such that $\sum _ { k = 1 } ^ { N } k a _ { k } = n$. If this set is finite, we denote by $p _ { n , N }$ its cardinality. We denote by $p _ { n }$ the final value of $\left( p _ { n , N } \right) _ { N \geq 1 }$.
Let $N \in \mathbf { N } ^ { * }$. Give a sequence $\left( a _ { n , N } \right) _ { n \in \mathbf { N } }$ such that
$$\forall z \in D , \frac { 1 } { 1 - z ^ { N } } = \sum _ { n = 0 } ^ { + \infty } a _ { n , N } z ^ { n }$$
Deduce, by induction, the formula
$$\forall N \in \mathbf { N } ^ { * } , \forall z \in D , \prod _ { k = 1 } ^ { N } \frac { 1 } { 1 - z ^ { k } } = \sum _ { n = 0 } ^ { + \infty } p _ { n , N } z ^ { n }$$

For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality. We denote by $p_n$ the final value of $(p_{n,N})_{N \geq 1}$.
Let $N \in \mathbf{N}^*$. Give a sequence $(a_{n,N})_{n \in \mathbf{N}}$ such that $$\forall z \in D, \frac{1}{1-z^N} = \sum_{n=0}^{+\infty} a_{n,N} z^n$$ Deduce, by induction, the formula $$\forall N \in \mathbf{N}^*, \forall z \in D, \prod_{k=1}^{N} \frac{1}{1-z^k} = \sum_{n=0}^{+\infty} p_{n,N} z^n$$

We consider the special case of a series of the form $f = I + F, F \in O_2$. We denote $G := f^\dagger - I, G \in O_2$. Assume that there exist $s > 0$ and $\alpha \in ]0,1[$ such that $\hat{F}(s) \leqslant \alpha s$. Show then that for all $d \geqslant 2, \widehat{[G]_d}((1-\alpha)s) \leqslant \alpha s$. Conclude that $$\hat{G}((1-\alpha)s) \leqslant \alpha s.$$

We have $P ( z ) = \sum _ { n = 0 } ^ { + \infty } p _ { n } z ^ { n }$ for all $z \in D$, where $p_n$ denotes the number of partitions of $n$.
Let $n \in \mathbf { N }$. Show that for all real $t > 0$,
$$p _ { n } = \frac { e ^ { n t } P \left( e ^ { - t } \right) } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta \tag{1}$$