grandes-ecoles 2024 Q20

grandes-ecoles · France · mines-ponts-maths1__psi Differential equations Entropy, Information, or Log-Sobolev Functional Analysis
Establish the following inequality
$$\operatorname{Ent}_{\varphi}(f) \leq \frac{1}{2} \int_{-\infty}^{+\infty} \frac{f^{\prime 2}(x)}{f(x)} \varphi(x) \mathrm{d}x$$
Establish the following inequality

$$\operatorname{Ent}_{\varphi}(f) \leq \frac{1}{2} \int_{-\infty}^{+\infty} \frac{f^{\prime 2}(x)}{f(x)} \varphi(x) \mathrm{d}x$$