Let $W$ be a random variable such that $\mathbb{P}(W > 0) = 1$ and $\mathbb{E}(W) = 1$. Is there an interpretation or motivation for the condition $$\mathbb{E}(W \log (W) ) < c$$ where $c \in (0,\infty)$ is a positive constant? Perhaps if we additionally assume that $W$ is absolutely continuous (with respect to the Lebesgue measure)?

If $W$ is a log-normal variable for example, i.e. $W = \exp(\sigma X - \sigma^2 / 2)$ where $X$ is a standard normal random variable, the condition $$\mathbb{E}(W \log (W) ) < c$$ is equivalent to $\sigma^2 < 2*c$. This is somehow a condition that 'most of the mass is concentrated around one'. Can this motivation/interpretation be generalized?