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 $x, y \in \mathbb{R}$. We set $z = \lambda x + (1-\lambda) y$. Prove that if $|y| \leq M$ then $\Psi(x) \leq \Psi_{M}(z)$. Similarly, prove that if $|x| \leq M$ then $\Psi(y) \leq \Psi_{M}(z)$.
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 $x, y \in \mathbb{R}$. We set $z = \lambda x + (1-\lambda) y$. Prove that if $|y| \leq M$ then $\Psi(x) \leq \Psi_{M}(z)$. Similarly, prove that if $|x| \leq M$ then $\Psi(y) \leq \Psi_{M}(z)$.