(a) For non-negative numbers $a, b, c$ and any positive real number $r$ prove the following inequality and state precisely when equality is achieved. $$a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r(c-a)(c-b) \geq 0$$ Hint: Assuming $a \geq b \geq c$ do algebra with just the first two terms. What about the third term? What if the assumption is not true?
(b) As a special case obtain an inequality with $a^4 + b^4 + c^4 + abc(a+b+c)$ on one side.
(c) Show that if $abc = 1$ for positive numbers $a, b, c$, then $$a^4 + b^4 + c^4 + a^3 + b^3 + c^3 + a + b + c \geq \frac{a^2+b^2}{c} + \frac{b^2+c^2}{a} + \frac{c^2+a^2}{b} + 3.$$

23. Solution: (1) Since $a^2 + b^2 \geq 2ab$, $b^2 + c^2 \geq 2bc$, $c^2 + a^2 \geq 2ac$, and $abc = 1$, we have $a^2 + b^2 + c^2 \geq ab + bc + ca = \frac{ab + bc + ca}{abc} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. Therefore $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq a^2 + b^2 + c^2$.
(2) Since $a, b, c$ are positive numbers and $abc = 1$, we have $(a+b)^3 + (b+c)^3 + (c+a)^3 \geq 3\sqrt[3]{(a+b)^3(b+c)^3(a+c)^3}$ $= 3(a+b)(b+c)(a+c)$ $\geq 3 \times (2\sqrt{ab}) \times (2\sqrt{bc}) \times (2\sqrt{ac})$ $= 24$. Therefore $(a+b)^3 + (b+c)^3 + (c+a)^3 \geq 24$.

[Elective 4-5: Inequalities] Let $a , b , c \in \mathbf { R } , a + b + c = 0 , a b c = 1$ .
(1) Prove: $a b + b c + c a < 0$;
(2) Let $\max \{ a , b , c \}$ denote the maximum value among $a , b , c$. Prove: $\max \{ a , b , c \} \geqslant \sqrt[3]{\frac{3}{2}}$.

[Elective 4-5: Inequalities] Given that $a , b , c$ are all positive numbers and $a ^ { 2 } + b ^ { 2 } + 4 c ^ { 2 } = 3$ , prove:
(1) $a + b + 2 c \leq 3$ ;
(2) If $b = 2 c$ , then $\frac { 1 } { a } + \frac { 1 } { b } + \frac { 4 } { c } \geq 3$ .

[Elective 4-5: Inequalities] (10 points) Given that $a$, $b$, $c$ are all positive numbers and $a ^ { 2 } + b ^ { 2 } + 4 c ^ { 2 } = 3$, prove that:
(1) $a + b + 2 c \leq 3$;
(2) If $b = 2 c$, then $\frac { 1 } { a } + \frac { 1 } { c } \geq 3$.

We seek to show that the inequality $|\sin(n\theta)| \leqslant n \sin(\theta)$ is satisfied for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2}\right]$.
a) Show that $\sin(n\theta) \leqslant n \sin(\theta)$ for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2n}\right]$.
b) Show that, for all $\theta \in \left[0, \frac{\pi}{2}\right]$, we have $\sin(\theta) \geqslant \frac{2}{\pi} \theta$.
c) Deduce that: $$\forall \theta \in \left[\frac{\pi}{2n}, \frac{\pi}{2}\right], \quad 1 \leqslant n \sin(\theta)$$
d) Conclude.
e) For which values of $\theta \in \left[0, \frac{\pi}{2}\right]$ do we have $|\sin(n\theta)| = n \sin(\theta)$?

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order.
Show that, for all $x \in [x_{n,1}, x_{n,n}]$, we have $$\sqrt{1 - x^2} \geq \frac{1}{n}$$

Let $P \in E_n$. Show that: $$\sup_{x \in [-1,1]} \left| P'(x) \right| \leq n^2 \sup_{x \in [-1,1]} |P(x)|$$ (One may use the trigonometric polynomial $T(\theta) = P(\cos(\theta))$.)

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Show that if $q_{/F} = 0$, then $\operatorname{dim}(F) \leq \frac{n}{2}$.

Let $\lambda$ be a real number in the interval $]0,1[$, and let $a$ and $b$ be two non-negative real numbers. Show that $$\lambda a + (1-\lambda) b \geq a^{\lambda} b^{1-\lambda}$$ (one may introduce a certain auxiliary function and justify its concavity). Moreover, show that for all real $u > 1$, $$(\lambda a + (1-\lambda) b)^{u} \leq \lambda a^{u} + (1-\lambda) b^{u}$$

Let $a$ and $b$ be two non-negative real numbers and $\lambda$ a real number in $]0,1[$. Show that $$(a+b)^{\lambda} \leq a^{\lambda} + b^{\lambda}$$

We set $\Psi(u) = \exp(-u^{2})$ for all real $u$. Prove that for all $x, y \in \mathbb{R}$, $$\Psi(\lambda x + (1-\lambda) y) \geq \Psi(x)^{\lambda} \Psi(y)^{1-\lambda}$$

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: $$\forall x, y \in \mathbb{R},\quad \chi_{n}(x)^{\lambda} \chi_{n}(y)^{1-\lambda} \leq \chi_{n+1}(\lambda x + (1-\lambda) y)$$

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Prove that for all $(j, k) \in \mathcal{I}$ and $(x, y) \in [0,1]^{2}$, we have $$|\theta_{j,k}(x) - \theta_{j,k}(y)| \leq 2^{j+1} |x - y|$$

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 $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$. Let $j$ be an integer, $1 \leqslant j \leqslant n$, and $\mathcal{V}$ be a vector subspace of $\mathbb{R}^{n}$ of dimension $j$. Show that $$\inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle \leqslant m_{j}.$$ (One may use questions $\mathbf{2c}$ and $\mathbf{3a}$, by choosing $\mathcal{U} = \mathcal{W}_{j}$.)

Let $\ell$ and $m$ be two $n$-tuples of real numbers. We write $$\ell \preccurlyeq m \quad \text{if and only if, for every integer } j,\, 1 \leqslant j \leqslant n, \quad \ell_{j} \leqslant m_{j}.$$ Show that for every matrix $M \in \mathcal{S}_{n}(\mathbb{R})$, $(0, \ldots, 0) \preccurlyeq s^{\downarrow}\left(\|M\| I_{n} - M\right)$.

Let $\ell$ and $m$ be two $n$-tuples of real numbers. We write $$\ell \preccurlyeq m \quad \text{if and only if, for every integer } j,\, 1 \leqslant j \leqslant n, \quad \ell_{j} \leqslant m_{j}.$$ Let $L, M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ and $\ell = s^{\downarrow}(L)$. Show that $$\max_{1 \leqslant j \leqslant n} \left|\ell_{j} - m_{j}\right| \leqslant \|L - M\|.$$

Let $x , y$ be strictly positive vectors of $\mathbb { R } ^ { n }$ and let $S , R$ be two sign diagonal matrices.
(a) Show that $$( S x \mid R y ) \leq ( x \mid y )$$ with equality if and only if $R = S$.
(b) Prove the uniqueness of $S$ in Broyden's theorem.
(c) Show that $$\| S x + R y \| \leq \| x + y \|$$ with equality if and only if $R = S$.

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
Show that $\alpha$ is positive or zero.

Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits a variance relative to $m$. Show that $fm$ is integrable. As a consequence, the real $$\operatorname { Var } _ { m } ( f ) = \int f ( x ) ^ { 2 } m ( x ) d x - \left( \int f ( x ) m ( x ) d x \right) ^ { 2 }$$ is well defined. Show that $\operatorname { Var } _ { m } ( f ) \geqslant 0$.

Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits an entropy relative to $m$. We consider the function $h : [ 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $h ( 0 ) = 0$ and for $x > 0$, $h ( x ) = x \ln ( x )$.
2a. Show that $f ^ { 2 } m$ is integrable. As a consequence, the real $$\operatorname { Ent } _ { m } ( f ) = \int h \left( f ( x ) ^ { 2 } \right) m ( x ) d x - h \left( \int f ( x ) ^ { 2 } m ( x ) d x \right)$$ is well defined.
2b. Let $a > 0$. Show that $$\forall x \geqslant 0 , \quad h ( x ) \geqslant ( x - a ) h ^ { \prime } ( a ) + h ( a ) ,$$ with strict inequality if $x \neq a$.
2c. Show that $\operatorname { Ent } _ { m } ( f ) \geqslant 0$. You may use the previous question with $a = \int f ( x ) ^ { 2 } m ( x ) d x$.
2d. We assume here that for all $x \in \mathbb { R } , m ( x ) > 0$. Characterize the functions $f$ such that $\operatorname { Ent } _ { m } ( f ) = 0$.

We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$
We decide to replace all $\rho_{k}$ by zeros. We thus replace the corresponding factors of $R_{N}$, $$\frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}},$$ by factors $X^{2}$. We thus obtain a new polynomial $S_{N}$ of the same degree as $R_{N}$.
Show that $0 \leqslant S_{N}(x) \leqslant R_{N}(x)$ for all $x \in [-1,1]$, then that $S_{N} \in B_{N}$.

We choose an even polynomial in $B_{N}$, denoted $R_{N}$, which has the factorisation $$R_{N}(X) = \prod_{j=1}^{r} \frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}} \prod_{k=1}^{s} \frac{X^{2} + \rho_{k}^{2}}{1 + \rho_{k}^{2}} \prod_{\ell=1}^{t} \frac{X^{2} - w_{\ell}^{2}}{1 - w_{\ell}^{2}} \cdot \frac{X^{2} - \overline{w_{\ell}}^{2}}{1 - \overline{w_{\ell}}^{2}}.$$ After replacing all $\rho_k$ by zeros we obtained $S_N$.
Similarly, in the list of $c_{j}$, we decide to replace those that do not belong to $[-1,1]$ by zeros. We thus replace the corresponding factors of $S_{N}$, $$\frac{X^{2} - c_{j}^{2}}{1 - c_{j}^{2}}$$ by factors $X^{2}$. We thus obtain a new polynomial $T_{N}$.
Show that $0 \leqslant T_{N}(x) \leqslant S_{N}(x)$ for all $x \in [-1,1]$, then that $T_{N} \in B_{N}$.

Let $p$ and $q$ be two strictly positive reals such that $\frac{1}{p} + \frac{1}{q} = 1$. Show that, for all non-negative reals $a$ and $b$,
$$ab \leqslant \frac{a^{p}}{p} + \frac{b^{q}}{q}$$
You may use the concavity of the logarithm.