Do the distributions with the following property belong to a particular class of distributions?: the cumulative distribution function (CDF) of a random variable with mean $\mu$ satisfies
$$F(x)\geq 1-e^{- x/\mu},$$
in other words, it is lower bounded by the CDF of an exponential random variable with the same mean $\mu$.
Does it have a name? I cannot think of well-known examples of distributions that satisfy this, do they have a special name?