Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
Show that the image set of the application $w$ defined on $]0,1[$ by $$\forall t \in ]0,1[, \quad w(t) = \lambda u(t) + (1-\lambda) v(t),$$ is equal to $\mathbb{R}$. Then prove that $w$ defines a change of variable from $]0,1[$ to $\mathbb{R}$. Using this and $\int_{-\infty}^{+\infty} h(w)\,dw$, show that $f$, $g$ and $h$ satisfy the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}.$$

Let $a < b$ be two real numbers and $f : [a,b] \rightarrow \mathbb{R}$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x_0 \in [a,b]$ where $f$ attains its maximum, we have $a < x_0 < b$, and $f''(x_0) \neq 0$.
We admit the identity $\int_{-\infty}^{+\infty} \exp(-x^2) \mathrm{d}x = \sqrt{\pi}$.
Under hypothesis (H), show the asymptotic equivalence, as $t \rightarrow +\infty$, $$\int_a^b e^{tf(x)} \mathrm{d}x \sim e^{tf(x_0)} \sqrt{\frac{2\pi}{t|f''(x_0)|}}$$

Let $a < b$ be two real numbers and $f : [ a , b ] \rightarrow \mathbb { R }$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x _ { 0 } \in [ a , b ]$ where $f$ attains its maximum, we have $a < x _ { 0 } < b$, and $f ^ { \prime \prime } \left( x _ { 0 } \right) \neq 0$.
Under hypothesis (H), show the asymptotic equivalence, as $t \rightarrow + \infty$, $$\int _ { a } ^ { b } e ^ { t f ( x ) } \mathrm { d } x \sim e ^ { t f \left( x _ { 0 } \right) } \sqrt { \frac { 2 \pi } { t \left| f ^ { \prime \prime } \left( x _ { 0 } \right) \right| } }$$

Let $r$ be the function defined by
$$\begin{aligned} r : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show that the function $r$ is of class $\mathcal { C } ^ { 1 }$ on $] - \pi ; \pi [$ and that:
$$\forall \theta \in ] - \pi ; \pi \left[ , \quad r ^ { \prime } ( \theta ) = - \mathrm { i } e ^ { \mathrm { i } \theta } \cdot \int _ { 0 } ^ { + \infty } \frac { t ^ { x } } { \left( 1 + t \mathrm { e } ^ { \mathrm { i } \theta } \right) ^ { 2 } } \mathrm {~d} t . \right.$$
Hint: let $\beta \in ] 0 ; \pi [$, show that for all $\theta \in [ - \beta ; \beta ]$ and $t \in [ 0 , + \infty [$, $\left| 1 + t e ^ { i \theta } \right| ^ { 2 } \geq \left| 1 + t e ^ { i \beta } \right| ^ { 2 } = ( t + \cos ( \beta ) ) ^ { 2 } + ( \sin ( \beta ) ) ^ { 2 }$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
Deduce that $$Z_n(h) = \sqrt{\frac{n}{2\mathrm{e}^{\beta}\pi\beta}} \int_{-\infty}^{+\infty} \left(\sum_{x \in \Lambda_n} \mathrm{e}^{(t+h)s_n(x)}\right) \mathrm{e}^{-\frac{nt^2}{2\beta}} \mathrm{~d}t$$

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Show that $$Z_n(h) = \sqrt{\frac{n}{2\mathrm{e}^{\beta}\pi\beta}} \int_{-\infty}^{+\infty} \mathrm{e}^{-nG_h(x)} \mathrm{d}x$$

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We denote $\widehat{G}_h : x \longmapsto G_h(x + u_h) - \min G_h$. Show that $$\psi_n(h) = -G_h(u_h) - \frac{1}{2n}\ln\left(2\mathrm{e}^{\beta}\pi\beta\right) + \frac{1}{n}\ln\left(\int_{-\infty}^{+\infty} \mathrm{e}^{-n\widehat{G}_h\left(\frac{t}{\sqrt{n}}\right)} \mathrm{d}t\right)$$