The infimum values of the probability functions for some infinitely divisible distributions motivated by Chvátal's theorem

Let $B(n,p)$ denote a binomial random variable with parameters $n$ and $p$. Chvátal’s theorem says that for any fixed $n\ge 2$, as $m$ ranges over $\{0,\dots ,n\}$, the probability $q_m:=P(B(n,m\mathrm{/}n)\le m)$ is the smallest when $m$ is closest to $\frac{2n}{3}$. Motivated by this theorem, in this paper we consider the infimum value of the probability $P(X\le \kappa E[X])$, where $\kappa$ is a positive real number, and $X$ is a random variable whose distribution belongs to some infinitely divisible distributions including the inverse Gaussian, log-normal, Gumbel and logistic distributions.