Approximating Moment of Sum of RVs

Given

• $X_i$ are independent random variables.
• $|X_i| < 1$
• $E[X_i] = 0$
• $X = \sum_i^n X_i$
• $var(X)=\sigma$

Prove:

$$E(X^p)^{1/p} = O(\sqrt{p}\sigma +p)$$ for all even p

Things I've tried:

First note, that all terms with more than p/2 terms are 0 (since $E[X_i] = 0$).

Furthermore, note that $\sum_i X^{4} \leq \sum_i X^2$

So this ends up being some way to count the various terms involving exactly $t$ variables. I don't know how to count this. What should I try?

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