Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Calculate $\int _ { 0 } ^ { \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } \sin t \, d t$. Deduce that $c _ { k } \leq \frac { k + 1 } { 4 }$.

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$ For $t, x \in \mathbb{R}_{+}^{*}$, we set $g(t,x) = t^{2} + x/t$.
a) Show that for $x \in \mathbb{R}_{+}^{*}$, the function $t \in \mathbb{R}_{+}^{*} \mapsto g(t,x)$ is convex and admits a unique minimum at $t = M(x)$, which we will determine. Calculate $g(M(x),x)$.
b) Let $m \in \mathbb{R}_{+}$, and $x \in \mathbb{R}_{+}^{*}$. Show using the inequality $M(x) \leq x^{1/3}$ that $$T_{m}(x) \leq \int_{0}^{x^{1/3}} t^{m} e^{-g(t,x)} dt + e^{-\frac{19}{10} x^{2/3}} \left(\int_{x^{1/3}}^{\infty} t^{m} e^{-\frac{t^{2}}{20}} dt\right)$$
c) Show that for every $\varepsilon > 0$ (and $m \in \mathbb{R}_{+}$), we can find $C > 0$ such that $$\forall t \geq 1, \quad t^{m} \leq C t e^{\varepsilon t^{2}}$$ Deduce that for every $\varepsilon > 0$, $$e^{-\frac{19}{10} x^{2/3}} \left(\int_{x^{1/3}}^{\infty} t^{m} e^{-\frac{t^{2}}{20}} dt\right) = O_{x \rightarrow \infty}\left(e^{-\left[\frac{39}{20} - \varepsilon\right] x^{2/3}}\right)$$
d) Show that $$T_{m}(x) = O_{x \rightarrow \infty}\left(x^{\frac{m+1}{3}} e^{-3(x/2)^{2/3}}\right)$$

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Show that for $x \in \mathbb{R}_{+}^{*}$, $$T_{-1}(x) \leq \int_{0}^{1} e^{-1/u^{2}} \frac{du}{u} + \int_{1}^{\infty} e^{-xu} \frac{du}{u}$$ Deduce that $T_{-1}(x) \leq 2$ for $x \geq 1$ and that $$T_{-1}(x) \leq 2 + \int_{x}^{1} e^{-w} \frac{dw}{w} \leq 2 - \ln x$$ if $0 < x \leq 1$.
b) Let $L \in [0,1]$, and $\rho \in C([0,L])$. We set $$[F(\rho)](x) = \int_{0}^{L} \rho(y) T_{-1}(|x-y|) dy$$ Show that $[F(\rho)](x)$ is well-defined for $x \in [0,L]$ and that $$\|F(\rho)\|_{\infty} \leq (4L + 2L|\ln L|) \|\rho\|_{\infty}.$$

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$. Throughout the rest of this question we assume $x > 1$ and $y > 1$. We denote $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$.
We denote $F _ { x , y }$ the antiderivative on $\mathbb { R } ^ { + }$ of $t \mapsto e ^ { - t } t ^ { x + y - 1 }$ which vanishes at 0. Show that $$\forall t \in \mathbb { R } ^ { + } , F _ { x , y } ( t ) \leqslant \Gamma ( x + y )$$

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$. We define $\psi ( x ) = \frac { \Gamma ^ { \prime } ( x ) } { \Gamma ( x ) }$ on $\mathbb{R}^{+*}$, and $\psi$ is increasing on $\mathbb{R}^{+*}$.
Let $n$ be an integer $\geqslant 2$ and $x$ a real $> - 1$. We set $p = E ( x ) + 1$, where $E ( x )$ denotes the integer part of $x$. Prove that $$0 \leqslant \psi ( n + x + 1 ) - \psi ( n ) \leqslant H _ { n + p } - H _ { n - 1 } \leqslant \frac { p + 1 } { n }$$

We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$ We denote $\rho _ { N } ( t ) = f ( t ) - \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu }$ the remainder of the asymptotic expansion of $f$.
We fix $\delta > 0$ and $\alpha \in \mathbb { R }$. Show that for all $x > 0$, the function $t \mapsto e ^ { - t / x } t ^ { \alpha }$ is integrable on $[ \delta , + \infty [$ and that for all $n \in \mathbb { N }$, we have: $$\int _ { \delta } ^ { + \infty } e ^ { - t / x } t ^ { \alpha } d t = o \left( x ^ { n } \right) \quad \text { when } x \rightarrow 0 ^ { + }$$ Deduce that for all $n \in \mathbb { N }$, $$\int _ { \delta } ^ { + \infty } e ^ { - t / x } \rho _ { N } ( t ) d t = o \left( x ^ { n } \right) \quad \text { when } x \rightarrow 0 ^ { + }$$

We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$ We denote $\rho _ { N } ( t ) = f ( t ) - \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu }$ the remainder of the asymptotic expansion of $f$.
We fix $\varepsilon > 0$. Show the existence of $\delta > 0$ and a constant $C ^ { \prime }$ independent of $\varepsilon$ and $\delta$ such that $$\forall x > 0 , \quad \left| \int _ { 0 } ^ { \delta } e ^ { - t / x } \rho _ { N } ( t ) d t \right| \leqslant C ^ { \prime } \varepsilon x ^ { ( N + \lambda ) / \mu }$$

Let $a$ and $b$ be two real numbers such that $0 < a < b$. Show that, for all $t > 0$ and all $x \in [a, b]$,
$$t^{x} \leqslant \max\left(t^{a}, t^{b}\right) \leqslant t^{a} + t^{b}$$

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Show that $$\left. \left. \forall (n, u) \in \mathbb{N}^{*} \times \right] 0, 1 \right], \quad \left| 1 - (\cos(\sqrt{2u/n}))^{n} \right| \leqslant u$$

We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let $$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$
Show that $J_{n} = \frac{\pi}{2}$ for $1 \leqslant n \leqslant 7$ and that $\left(J_{n}\right)_{n \geqslant 7}$ is strictly increasing.

From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$. We are also given an infinitely differentiable function $g : [0,1] \rightarrow \mathbb{R}$.
We admit that there exist real numbers $d, d' \in \mathbb{R}$ such that, as $a \rightarrow +\infty$, $$\int_0^a \cos(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4} + \frac{d}{a} \sin(a^2) + \frac{d'}{a^3} \cos(a^2) + O\left(\frac{1}{a^5}\right)$$
Show that, as $t \rightarrow +\infty$, $$\int_{x_0}^1 g(x) \sin(tf(x)) \mathrm{d}x = g(x_0) \int_{x_0}^1 \sin(tf(x)) \mathrm{d}x + O\left(\frac{1}{t}\right)$$

From now on, $f$ denotes an infinitely differentiable function from $[ 0,1 ]$ to $\mathbb { R }$. We assume that there exists a unique point $x _ { 0 } \in \left[ 0,1 \left[ \right. \right.$ where $f ^ { \prime }$ vanishes. We also assume that $f ^ { \prime \prime } \left( x _ { 0 } \right) > 0$. We are also given an infinitely differentiable function $g : [ 0,1 ] \rightarrow \mathbb { R }$.
Show that we have, as $t \rightarrow + \infty$, $$\int _ { x _ { 0 } } ^ { 1 } g ( x ) \sin ( t f ( x ) ) \mathrm { d } x = g \left( x _ { 0 } \right) \int _ { x _ { 0 } } ^ { 1 } \sin ( t f ( x ) ) \mathrm { d } x + O \left( \frac { 1 } { t } \right)$$

For $k \in \mathbf { N } ^ { * }$ and $t \in \mathbf { R } _ { + }$, we set
$$u _ { k } ( t ) = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { t q ( u ) } { e ^ { t u } - 1 } \mathrm {~d} u \quad \text { if } t > 0 , \quad \text { and } \quad u _ { k } ( t ) = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { q ( u ) } { u } \mathrm {~d} u \quad \text { if } t = 0$$
where $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$.
Let $t \in \mathbf { R } _ { + } ^ { * }$. Show successively that $\left| u _ { k } ( t ) \right| = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { t | q ( u ) | } { e ^ { t u } - 1 } \mathrm {~d} u$, then $u _ { k } ( t ) = ( - 1 ) ^ { k } \left| u _ { k } ( t ) \right|$ for all integer $k \geq 1$, and finally establish that
$$\forall n \in \mathbf { N } ^ { * } , \left| \sum _ { k = n } ^ { + \infty } u _ { k } ( t ) \right| \leq \frac { 1 } { 2 n } .$$
We admit in what follows that this bound also holds for $t = 0$.

For $k \in \mathbf{N}^*$ and $t \in \mathbf{R}_+$, we set $$u_k(t) = \int_{k/2}^{(k+1)/2} \frac{tq(u)}{e^{tu}-1} \mathrm{du} \text{ if } t > 0 \text{, and } u_k(t) = \int_{k/2}^{(k+1)/2} \frac{q(u)}{u} \mathrm{du} \text{ if } t = 0.$$ Let $t \in \mathbf{R}_+$. Show successively that $|u_k(t)| = \int_{k/2}^{(k+1)/2} \frac{t|q(u)|}{e^{tu}-1} du$ then $u_k(t) = (-1)^k |u_k(t)|$ for all integer $k \geq 1$, and finally establish that $$\forall n \in \mathbf{N}^*, \left|\sum_{k=n}^{+\infty} u_k(t)\right| \leq \frac{1}{2n}.$$

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. It has been shown that $\left| U ( f ) ^ { \prime } ( x ) \right| \leqslant \| f \| \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } }$ and $\lim_{x\to 0} U(f)(x) = 0$. Deduce from the above that $U$ is an endomorphism of $E$ and that, for all $f \in E$ and all $x > 0$, $$| U ( f ) ( x ) | \leqslant 4 \| f \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$$

We assume that $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ is a decreasing sequence of strictly positive real numbers. We denote by $f$ the step function which, for all $k \in \mathbb { N } ^ { * }$, equals $a _ { k }$ on the interval $[ k - 1 , k [$.
Prove that, for all $k$ in $\mathbb { N } ^ { * }$,
$$\int _ { k - 1 } ^ { k } \exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right) \mathrm { d } x \geqslant \exp \left( \frac { 1 } { k } \sum _ { i = 1 } ^ { k } \ln \left( a _ { i } \right) \right)$$
You may use the previous question.