jasperE3 wrote: Prove that there is no polynomial $P(x)$ with real coefficients that satisfies $$P'(x)P''(x)>P(x)P'''(x)\qquad\text{for all }x\in\mathbb R.$$Is this statement true for all of the thrice differentiable real functions? Let $n$ be degree of $P(x)$ If $n\le 1$, then $P'P''-PP'''=0$ does not fit If $n\ge 2$ and highest degree summand of $P(x)$ is $ax^n$, then highest degree summand of $P'P''-PP'''$ is $2a^2n(n-1)x^{2n-3}$ And so $\lim_{x\to-\infty}(P'P''-PP''')=-\infty$. Hence the claim jasperE3 wrote: Is this statement true for all of the thrice differentiable real functions? No, this is not true for any thrice differentiable function. Choose for example the funny $f(x)=e^{e^{-x}}$