McDiarmid's Inequality basically says the following:

Let $X_1, X_2, X_3, \ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the $i$-th variable changes the value of $f$ by at most $c_i$, then
$$\Pr(f > E[f]+t), \Pr(f < E[f] -t) \leq \exp\left(\frac{-2t^2}{\sum_i c_i^2}\right)$$
Is there a known multiplicative version of this inequality, i.e. we have $\Pr(f > (1+\epsilon)E[f])$ on the left hand side.

The Chernoff-Hoeffding Bounds actually have the corresponding two versions:
$$\Pr[X > E[X] +t] \leq \exp(-2t^2/n)$$
$$\Pr[X > (1+\epsilon)E[X]] < \exp(-\epsilon^2E[X]/3).$$

I am wondering anyone has thought about it for McDiarmid's Inequality. Is it possible to derive the same multiplicative version just by going though the proofs of both of them?