By using the Cauchy-Schwarz inequality, show that $$\forall t \in \mathbf{R}_+^*, \quad -S'(t) \leq \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} P_t\!\left(\frac{f'^2}{f}\right)(x)\,\varphi(x)\,\mathrm{d}x.$$
By using the Cauchy-Schwarz inequality, show that
$$\forall t \in \mathbf{R}_+^*, \quad -S'(t) \leq \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} P_t\!\left(\frac{f'^2}{f}\right)(x)\,\varphi(x)\,\mathrm{d}x.$$