Let $X_1, X_2, \cdots, X_n$ be $n$ independent varaible which are between $0$ and $1$.
Then,
$Pr[X \ge (1 + \delta)U] \lt (\frac{e^{\delta}}{(1 + \delta)^{1 + \delta}})^U$,
$Pr[X \le (1 - \delta)L] \lt (\frac{e^{-\delta}}{(1 - \delta)^{1 - \delta}})^L$.
For $X = \sum\limits_{i=1}^n X_i$, $\mu = E[X]$, $L \le \mu \le U$ and $\delta > 0$.
If $0 \le \delta < 1$ then
$Pr[X \ge (1 + \delta)U] < (\frac{e^{\delta}}{(1 + \delta)^{1 + \delta}})^U < e^{-U\delta^2/3}$,
$Pr[X \le (1 - \delta)L] < (\frac{e^{-\delta}}{(1 - \delta)^{1 - \delta}})^L < e^{-L\delta^2/2}$
Proof will be updated later.