Tight bounds on Poisson tails with applications to Szász-Mirakyan operators

We obtain tight bounds on Poisson tails which are easy to handle. A short proof based on the median of the gamma distribution is given. Numerical comparisons with other known estimates are made. As an application, we consider the rates of pointwise convergence for $S_{n}f(x)-f(x)$, as $n\to \mathrm{\infty }$, where $S_n$ is the Szász-Mirakyan operator and $x$ belongs to an open interval $I$. If the function $f$ is affine on $I$, the rate of convergence is exponential. This property is no longer true at the boundary of $I$