Let $Q$ is a matrix with every element $0<q_{i,j}<1$ and each row sum of $Q$ being 1. Let $I$ be an identity matrix. Then, it is clear that $\|Q-I \|_{\infty} \le \|Q \|_{\infty} + \|I \|_{\infty} = 2$. Now I would like to find an upper bound on the norm $\|\exp(Q-I)\|_{\infty}$, where $\exp(\cdot)$ is defined as $$ \exp(X) = I+X+\frac{X^2}{2!}+\frac{X^3}{3!}+... $$
My initial idea is as follows:
$$ \|\exp(Q-I)\|_{\infty} \le \exp(\|Q-I\|_{\infty}) \le \exp(2) $$
However, having tested this bound using some numerical values of $Q$, I find this bound a bit loose. I notice that, for the numerical values that I tested, the bound $\|\exp(Q-I)\|_{\infty} \le 2$ holds, but I do want to know how to mathematically prove it.
Ang suggestion is much appreciated.