An $\ell_p$ inequality for discrete martingales

Question

If $Y_n=\sum_{i=1}^n X_i$ is a martingale, where $X_i$ is a martingale difference sequence, $\mathbb{E}[X_n\mid \mathcal{F}_{n-1}]=0$ for all $n$, we know that $$ \mathbb{E}\big[Y_n^2-Y_{n-1}^2\big]=\mathbb{E}X_n^2,$$ by using the simple fact that $Y_n^2=Y_{n-1}^2+2X_nY_{n-1}+X_n^2$, where the cross term vanishes since $Y_{n-1}$ is $\mathcal F_{n-1}$ measurable and $X_i$ centred, that is, $$ \mathbb{E}[X_nY_{n-1}]=\mathbb{E}\big[Y_{n-1}\mathbb{E}[X_n\mid \mathcal{F}_{n-1}]\big] =0.$$ A similar property, but now as an inequality, holds if we replace the square with the absolute value, $$ \mathbb{E}\big[|Y_n|-|Y_{n-1}|\big]\le\mathbb{E}|X_n|.$$ Does something analogous hold for other powers? Namely, something along the lines of $$ \mathbb{E}\big[|Y_n|^r-|Y_{n-1}|^r\big]\le C\mathbb{E}|X_n|^r,$$ for $1<r<2$ and some $C>0$?

Answer 1

The question was affirmatively answered on mathoverflow. Specifically, applying inequality (1.1) in this paper (DOI: 10.15352/afa/06-4-1) with $n=2$ yields $$ \mathbb{E}[|Y_n|^r] \le \mathbb{E}[|Y_{n-1}|^r]+C_r\mathbb{E}[|X_n|^r], $$ where $1\le C_r\le 2$. The paper states that an inequality of this types holds not just for the power function $\lvert \cdot\lvert^r$, but for a certain class of generalised moment functions, where the corresponding constant depends on the moment function.