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 $lambda$.