Let $n \in \mathbb{N}^{*}$. We denote by $\chi_{n} : \mathbb{R} \rightarrow \mathbb{R}$ the continuous function that equals 1 on $[-n, n]$, equals 0 on $]-\infty, -n-1] \cup [n+1, +\infty[$ and is affine on each of the two intervals $[-n-1, -n]$ and $[n, n+1]$.
Show that: $$\forall x, y \in \mathbb{R},\quad \chi_{n}(x)^{\lambda} \chi_{n}(y)^{1-\lambda} \leq \chi_{n+1}(\lambda x + (1-\lambda) y)$$

Show that the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}$$ is satisfied (if you choose to use the dominated convergence theorem then carefully verify that its conditions of validity are satisfied).

Let $N : \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a norm on the vector space $\mathbb{R}^{n}$. Prove that the application defined by $$\forall x \in \mathbb{R}^{n}, \quad f(x) = \exp\left(-N(x)^{2}\right),$$ is continuous and log-concave on $\mathbb{R}^{n}$. (One may observe that the function $u \mapsto u^{2}$ is convex on $\mathbb{R}_{+}$).

Let $\lambda \in ]0,1[$ and $f, g, h$ be functions from $\mathbb{R}^{2}$ to $\mathbb{R}_{+}$ that are continuous with bounded support and such that $$\forall X \in \mathbb{R}^{2}, \forall Y \in \mathbb{R}^{2}, \quad h(\lambda X + (1-\lambda) Y) \geq f(X)^{\lambda} g(Y)^{1-\lambda}$$ Show that $$\iint_{\mathbb{R}^{2}} h(x,y)\,dx\,dy \geq \left(\iint_{\mathbb{R}^{2}} f(x,y)\,dx\,dy\right)^{\lambda} \left(\iint_{\mathbb{R}^{2}} g(x,y)\,dx\,dy\right)^{1-\lambda}.$$

We consider a rectangle $]a,b[ \times ]c,d[$ of the plane $\mathbb{R}^{2}$, with $a < b$ and $c < d$. Calculate the real number $V(]a,b[ \times ]c,d[)$. What does it represent? (One may use functions of the type $$(x,y) \mapsto f(x,y) = \phi(x)\varphi(y)$$ where $\phi$ and $\varphi$ are well-chosen continuous and piecewise affine functions).

Let $\mathcal{A}$ and $\mathcal{B}$ be two open bounded non-empty subsets of $\mathbb{R}^{2}$ and $\lambda \in ]0,1[$. Verify that $\lambda\mathcal{A} + (1-\lambda)\mathcal{B}$ is an open bounded subset of $\mathbb{R}^{2}$. Then show that $$V(\lambda\mathcal{A} + (1-\lambda)\mathcal{B}) \geq V(\mathcal{A})^{\lambda} V(\mathcal{B})^{1-\lambda}$$ To prove this inequality, you will use the following admitted result. For all $f \in C(\mathcal{A})$ and $g \in C(\mathcal{B})$, the function $h$ determined by: $$\forall Z \in \mathbb{R}^{2},\quad h(Z) = \sup\left\{f(X)^{\lambda} g(Y)^{1-\lambda} \,/\, X, Y \in \mathbb{R}^{2},\, Z = \lambda X + (1-\lambda) Y\right\}$$ defines a continuous function on $\mathbb{R}^{2}$.

Let $x$ be a strictly positive real number, $\beta$ a real number such that $0 < \beta < 1$.
Prove that: $x ^ { \beta } \leqslant \beta x + 1 - \beta$.

Show that $\forall x \in \mathbb { R }$ and $\forall n \in \mathbb { N } ^ { * }$ $$\prod _ { k = 1 } ^ { n - 1 } \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right) = \left( \sum _ { k = 0 } ^ { n - 1 } x ^ { k } \right) ^ { 2 }$$

Show that $V$ is a vector subspace of $\mathbf{C}^{\mathbf{Z}}$. Given $f \in \mathbf{C}^{\mathbf{Z}}$, we define $E(f) \in \mathbf{C}^{\mathbf{Z}}$ by $E(f)(k) = f(k+1), k \in \mathbf{Z}$.

Show that $E \in \mathcal{L}(\mathbf{C}^{\mathbf{Z}})$ and that $V$ is stable under $E$.

We assume that the conditions of questions 4 and 5 are satisfied and that $\lambda(0) = 0, \mu(0) = 1$.
7a. Show that $F \in \mathrm{GL}(V)$.
7b. Show that $E$ and $F$ are not of finite order in the group $\mathrm{GL}(V)$.
7c. Calculate the kernel of $H$ and show that $H^r \neq \operatorname{Id}_V$ for $r \geq 1$.

Let $s \in ]0,1[$. Show that if $a, b \geq 0$, then $a^{s} + b^{s} \leq 2^{1-s}(a+b)^{s}$.

Let $s \in ]0,1[$. Show that if $f \in \Gamma^{s}(x_{0}) \cap \mathcal{C}_{0}$, then there exists a real number $c_{1} > 0$, such that for all $(j, k) \in \mathcal{I}$, we have $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s} .$$ Recall that $c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2}$.

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We recall that $\widetilde{k}_{j}(x)$ is the integer part of $2^{j} x$. We set $$W_{j} = \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x) - \theta_{j,k}(x_{0})|$$ Show that $$W_{j} \leq (|c_{j,\widetilde{k}_{j}(x)}(f)| + |c_{j,\widetilde{k}_{j}(x_{0})}(f)|) 2^{j+1} |x - x_{0}|$$

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$.
Show that for $j \leq n_{0}$, we have $$W_{j} \leq 4 c_{1} 2^{(1-s)j} 3^{s} |x - x_{0}|$$

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$.
Deduce that, by setting $c_{2} = 8(2^{1-s} - 1)^{-1} (3/2)^{s} c_{1}$, $$\sum_{j=0}^{n_{0}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x) - \theta_{j,k}(x_{0})| \leq c_{2} |x - x_{0}|^{s}$$

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We recall that $\widetilde{k}_{j}(x_{0})$ is the integer part of $2^{j} x_{0}$.
Show that for all $j \in \mathbf{N}$, $|c_{j,\widetilde{k}_{j}(x_{0})}(f)| \leq 2^{s(1-j)} c_{1}$. Deduce, by setting $c_{3} = (1 - 2^{-s})^{-1} 2^{s} c_{1}$, $$\sum_{j=n_{0}+1}^{+\infty} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x_{0})| \leq c_{3} |x - x_{0}|^{s}$$

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$. We suppose that $\|f\|_{\infty} = 1$.
Show that there exists a unique $n_{1} \in \mathbf{N}$ such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$, and $n_{1}$ the unique natural integer such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.
Show that for all $n \geq n_{1}$, we have $$\|f - S_{n} f\|_{\infty} \leq 2^{s+1} |x - x_{0}|^{s}$$ One may use the results of questions 9a and 9c.

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$, and $n_{1}$ the unique natural integer such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.
Show that when $n_{0} < n_{1}$, we have $$\sum_{j=n_{0}+1}^{n_{1}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x)| \leq c_{1} 3^{s} (n_{1} - n_{0}) |x - x_{0}|^{s} .$$

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Let $\eta = \xi^{-1} : I \rightarrow \mathbb{R}$ be the inverse function of the bijection $\xi : \mathbb{R} \rightarrow I$. Show that where it is defined $$(\eta(xy))^2 = \eta\left(x^2\right)\eta\left(y^2\right)$$

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Deduce in the general case that, if $\xi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function satisfying condition (V.1), then it is odd and its restriction to $\mathbb{R}_+^*$ is of the form $x \mapsto Cx^\beta$, with $C \neq 0$ and $\beta > 0$.

For $\lambda \in \mathbb{R}$, let $A_\lambda \in \mathcal{M}_d(\mathbb{R})$ be the matrix having only 1's off the diagonal and only $\lambda$ on the diagonal.
Deduce all continuous functions $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Show that for all $x$ in $\mathbb{R}^n$, and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $f_{i,j,k}(x) = -f_{i,k,j}(x)$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Deduce that, for all $x$ in $\mathbb{R}^n$ and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, we have $f_{i,j,k}(x) = 0$.