Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. We denote $\Lambda = \min(\lambda, 1-\lambda)$, $\Theta = \max(\lambda, 1-\lambda)$ and $\widehat{M} = M\max(\lambda, 1-\lambda)$. For each real $u$ we set: $$\Psi_{M}(u) = \begin{cases} \exp\left(-\frac{1}{\Theta^{2}}(|u| - \widehat{M})^{2}\right), & \text{if } |u| > \widehat{M} \\ 1, & \text{if } |u| \leq \widehat{M} \end{cases}$$ Let $\epsilon \in ]0,1[$, $f_{\epsilon} = f + \epsilon\Psi$ and $g_{\epsilon} = g + \epsilon\Psi$. Show that $$\forall x, y \in \mathbb{R}, \quad f_{\epsilon}(x)^{\lambda} g_{\epsilon}(y)^{1-\lambda} \leq h(z) + \epsilon^{\Lambda}\left(\|f\|_{\infty}^{\lambda} + \|g\|_{\infty}^{1-\lambda}\right)\left(\Psi_{M}(z)\right)^{\Lambda} + \epsilon\Psi(z)$$ where $z = \lambda x + (1-\lambda) y$. One should begin by applying the inequality from question 2, then the two preceding questions. (We recall that $f(x) = 0$ if $|x| > M$ and that $g(y) = 0$ if $|y| > M$).
Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. We denote $\Lambda = \min(\lambda, 1-\lambda)$, $\Theta = \max(\lambda, 1-\lambda)$ and $\widehat{M} = M\max(\lambda, 1-\lambda)$. For each real $u$ we set:
$$\Psi_{M}(u) = \begin{cases} \exp\left(-\frac{1}{\Theta^{2}}(|u| - \widehat{M})^{2}\right), & \text{if } |u| > \widehat{M} \\ 1, & \text{if } |u| \leq \widehat{M} \end{cases}$$
Let $\epsilon \in ]0,1[$, $f_{\epsilon} = f + \epsilon\Psi$ and $g_{\epsilon} = g + \epsilon\Psi$. Show that
$$\forall x, y \in \mathbb{R}, \quad f_{\epsilon}(x)^{\lambda} g_{\epsilon}(y)^{1-\lambda} \leq h(z) + \epsilon^{\Lambda}\left(\|f\|_{\infty}^{\lambda} + \|g\|_{\infty}^{1-\lambda}\right)\left(\Psi_{M}(z)\right)^{\Lambda} + \epsilon\Psi(z)$$
where $z = \lambda x + (1-\lambda) y$. One should begin by applying the inequality from question 2, then the two preceding questions. (We recall that $f(x) = 0$ if $|x| > M$ and that $g(y) = 0$ if $|y| > M$).