Are derivatives of a characteristic function bounded?

Question

Let $X$ be a real valued random variable with cdf $F(x)$ and characteristic function $\varphi(t)$, and suppose that $E[|X|^n]<\infty$ for some $n$. Then we know $$\varphi^{(k)}(t)=i^k\int_{-\infty}^\infty x^ke^{itx}dF(x),$$ incidentally $i^kE[X^k]=\varphi^{(k)}(0)$. My question is, are derivatives of $\varphi$ bounded? So that for each $k$ there exists some $M_k$ such that $$|\varphi^{(k)}(t)|\leq M_k.$$ At least for $\varphi$ itself we know that $$|\varphi(t)|\leq 1=\varphi(0),$$ can similar conclusions be drawn for $\varphi^{(k)}(t)$?

Answer 1

Starting from the formula $$\varphi^{(k)}(t)=i^k\int_{-\infty}^\infty x^ke^{itx}\mathrm dF(x),$$ we have for each real number $t$ and each integer $k\lt n$: $$\left|\varphi^{(k)}(t)\right|=\left|\int_{-\infty}^\infty x^ke^{itx}\mathrm dF(x)\right|\leqslant \int_{-\infty}^{+\infty}\left|x\right|^k\mathrm dF(x)=\mathbb E\left[\left|X\right|^k\right].$$