Is there a tighter version of McDiarmid inequality $$ \Pr\left(g(X_1,\dotsc,X_n)) - \mathbb{E}[g(X_1,\dotsc,X_n)] > t\right) \le 2e^{2t^2/\sum_{i=1}^n(b_i-c_i)^2} $$ I'm referring specifically to the coefficent and the form of the coefficient.
We often see the function $\phi(x) = (1+x)\ln(1+x) - x$ appearing in the exponent of tighter forms of other large deviation bounds, but I can't find any reference presenting McDiarmid using this function or others.