Expectation of function and its derivatives

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I was wondering if anyone has an idea of what could be said about $f$ given that it satisfies the following inequality: $$\mathbb{E}\left[ \frac{1}{\sigma^2}f(x)^2+f'(x)^2+2f(x)f''(x) \right] \geq 0$$ $$x \sim \mathbb{N}(\mu,\sigma^2)$$ Or if I can brake this two smaller rules?