My friend sent me the following Exercice for solving :
Exercice: Write the generator of normal law $N(0,1)$ using the following inequality :
$\forall x\in \mathbb{R}: \dfrac{1}{\sqrt{2\pi}}\exp(-x^2/2)\leq \sqrt{\dfrac{2\pi}{e}}\dfrac{1}{\pi}\dfrac{1}{x^2+1}\tag{1}$
Attempt: I have tried to find moment generatrice function by assumption that I have a continious probability density function such that I tried to evaluate $M_X(t) =\displaystyle \int_{-\infty}^{\infty}\exp(tx) f(x) dx $ with $f(x)=\exp(-x^2/2)$ then i multiplied both side of inequality $(1)$ by $\exp(tx) $ the inequality become : $\dfrac{1}{\sqrt{2\pi}}\exp(tx)\exp(-x^2/2)\leq \sqrt{\dfrac{2\pi}{e}}\dfrac{1}{\pi}\dfrac{\exp(tx)}{x^2+1}\tag{2}$, integrate both side over $(-\infty,\infty)$ then assume I have got closed form of RHS and LHS of inequality $(2)$ after integration , Does this mean that I have got something relate to generator of Normal law $N(0,1)$ ? and if no then how I can get that generator using inequality $(1)$?