Inequality for increments of $r$th absolute moments of martingales, $1<r<2$

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 $$ \mathbb{E}\big[Y_n^2-Y_{n-1}^2\big]=\mathbb{E}X_n^2.$$ 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 $C>0$?

I guess if the distribution of $X_n$ given $Y_{n-1}$ were symmetric about zero, then this would be a direct consequence of von Bahr-Essen bounds (Inequalities for the rth Absolute Moment of a Sum of Random Variables, $1 \le r\le 2$, The Annals of Mathematical Statistics, 36), with $C=1$. Does it also hold under weaker assumptions? The von Bahr-Essen paper states the first equation (equality in case of $r=2$) as a special case.

Answer 1

Yes, this inequality, with the best possible $C$ ($\le 2$), was proved in this paper; see e.g. inequality (1.11) there.

Indeed, that inequality implies that $$E\Big|\sum_{i=1}^m U_i\Big|^r\le E|U_1|^r+C_r\sum_{i=2}^m E|U_i|^r,$$ where the $U_i$'s are martingale differences and $C_r\in[1,2]$. Taking here $m=2$, $U_1:=Y_{n-1}$, and $U_2:=X_n$, we get $$E|Y_n|^r\le E|Y_{n-1}|^r+C_rE|X_n|^r,$$ as desired.