Is there any result for upper bounding the tail of a sum of r.v.s by another tail (with a different threshold)?

Question

Suppose $X_i$s are independent random variables. We can make assumptions about $X_i$, e.g., $X_i\in [0,1]$. Let $X=\sum_i X_i$ and $u=E[X]$. Is there any result of the following type (relating one tail probability with another tail)? E.g.,

For some constant $1<L<L'$, $$\mathbb P[X > L' u] \leq c \cdot\mathbb P[X > Lu]$$ for some constant $c<1$ (which may depend on $L$ and $L'$).