My question comes from exercise 3.21 in this monograph. Specifically, let $f$ be a real-valued function defined on the set of non-negative integers and denote its "discrete derivative" by $Df(x) = f(x+1)-f(x)$. Let $X$ be a Poisson random variable with parameter $\mathbb E[X] = \mu$. Prove that $$\mathrm{Var}(f(X)) \leq \mu\,\mathbb E\left[(Df(X))^2\right].$$ The hint is to use the Efron-Stein inequality and the infinite divisibility of the Poisson distribution.
My attempt: the infinite divisibility of the Poisson distribution means that we can write $X := \sum_{i=1}^n X_i$, where $X_i$'s are i.i.d. with Poisson distribution $\mathrm{Poisson}(\mu/n)$. However, I do not see how the Efron-Stein inequality (Theorem 3.1) $$\mathrm{Var}(f(X)) \leq \sum_{i=1}^n \mathbb{E}\left[\left(f(X) - \mathbb{E}^{(i)}f(X)\right)^2\right] $$ will lead us to the advertised inequality. Here the conditional expectation operator $\mathbb{E}^{(i)}$ means that we are taking the expectation of $f(X)$ with respect to $X_i$ (while keeping $X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_n$ fixed). Any help will be greatly appreciated!