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I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$: $$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$ Numerically, I see that ...

Let $V \in C^2(\mathbb{R}^d; \mathbb{R})$ a (strictly) convex function with $ \int_{\mathbb{R}^d} \mathrm{e}^{-V(x)} \, dx = 1.$ I am trying to show that $$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...