Suppose that $X_1,\ldots,X_n$ are independent random variables with $\operatorname E X_k=0$ and $\operatorname E |X_k|^p<\infty$ with $1<p<2$ for each $1\le k\le n$. I am interested in the inequalities that establish a lower bound for the $p$-th absolute moment of $S_n=\sum_{k=1}^nX_k$ in terms of the $p$-th absolute moments of $X_1,\ldots,X_n$.

I was able to find an upper bound for $E|S_n|^p$. von Bahr and Esseen (1965) among other results established that $$ \operatorname E|S_n|^p\le2\sum_{k=1}^n\operatorname E|X_k|^p. $$ But I can't seem to find an inequality that establishes a lower bound for $\operatorname E|S_n|^p$. My questions are as follow:

Are there any inequalities that establish a lower bound for $\operatorname E|S_n|^p$ in terms of the $p$-th absolute moments of $X_1,\ldots,X_n$? Is it true that $\operatorname E|S_n|^p\ge C\sum_{k=1}^n\operatorname E|X_k|^p$ with some positive constant $C$?

Any help is much appreciated!

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