When does $k$-th central moment relate to $k$-th power of expectation?

Question

Given a bounded random variable $X$. When exactly is the $k$-th central moment $ E (X-\mu)^k $ upper bounded by the $k$-th power of the expectation $ \mu^k$? This seems to be the case for bounded, exponentially distributed random variables. Is there a more general condition or even a equivalent characterisation when that holds?