An application of taylor formula

Question

Let $\phi\in \mathcal{S}(\mathbb{R}^n)$, i.e. $\phi$ is a Schwartz function, is true that: $$|2\phi(x)-\phi(x+y)-\phi(x-y)|\leq |D^2\phi(x)||y|^2,\quad\forall y\in\mathbb{R}^n???$$ I have no idea on how to proceed, any help is appreciated.

Answer 1

No, this inequality is not true. For a counterexample, let $n=1$ be the dimension and consider the function $\phi(x) = (x^2+x^4)\rho(x)$, where $\rho(x)$ is some smooth compactly-supported bump function with $\rho \equiv 1$ on $(-1,1)$. Then clearly $\phi\in\cal S(\Bbb R)$ since it is smooth and compactly-supported.

Now, in a neighborhood of $0$ we have $\phi''(x) = 2+12x^2$, meaning $\phi''(0)=2$. Since $\phi(0)=0$ as well, your inequality at $x=0$ becomes $$|-\phi(y)-\phi(-y)|\leq 2|y|^2.$$ On the other hand, for $y$ small we can compute the left-hand side to be $2(y^2+y^4)$. Since this exceeds $2y^2$ for $y\neq 0$, we conclude your inequality is false.