The density ratio of Poisson binomial versus Poisson distributions

Let $b(x)$ be the probability that a sum of independent Bernoulli random variables with parameters $p_1, p_2, p_3, \ldots \in [0,1)$ equals $x$, where $\lambda := p_1 + p_2 + p_3 + \cdots$ is finite. We prove two inequalities for the maximal ratio $b(x)/\pi_\lambda(x)$, where $\pi_\lambda$ is the weight function of the Poisson distribution with parameter $\lambda$.