On the Chvatal-Janson conjecture

In a recent paper, Svante Janson has considered a conjecture suggested by Va\v{s}ek Chv\a'atal dealing with the probability that a binomial random variable with parameters $n$ and $m/n$ - where $m$ is an integer - exceeds its expectation $m$. Albeit Janson has provided a proof of this conjecture for large $n$, we show that the result actually holds for each $n\geq 2$.