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 $g(x) > 0$. We denote $F = \int_{-\infty}^{+\infty} f(x)\,dx$ and $G = \int_{-\infty}^{+\infty} g(x)\,dx$. Show that for all $t$ in the interval $]0,1[$ there exists a unique real number denoted $u(t)$ and a unique real number denoted $v(t)$ such that $$\frac{1}{F} \int_{-\infty}^{u(t)} f(x)\,dx = t, \quad \frac{1}{G} \int_{-\infty}^{v(t)} g(x)\,dx = t$$ (One may study the variations of the function: $u \mapsto \frac{1}{F} \int_{-\infty}^{u} f(x)\,dx$).
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 $g(x) > 0$.
We denote $F = \int_{-\infty}^{+\infty} f(x)\,dx$ and $G = \int_{-\infty}^{+\infty} g(x)\,dx$. Show that for all $t$ in the interval $]0,1[$ there exists a unique real number denoted $u(t)$ and a unique real number denoted $v(t)$ such that
$$\frac{1}{F} \int_{-\infty}^{u(t)} f(x)\,dx = t, \quad \frac{1}{G} \int_{-\infty}^{v(t)} g(x)\,dx = t$$
(One may study the variations of the function: $u \mapsto \frac{1}{F} \int_{-\infty}^{u} f(x)\,dx$).