On the conditions of the uniform integrability of the continuous nonnegative martingales

The problem of the uniform integrability of martingales
$$ Z_t=\exp\{\mu_t-1/2\langle\mu\rangle_t\} $$
is considered, where $\mu_t$ is a continuous martingale and $\langle\mu\rangle_t$ is its increasing process. It is shown that the condition
$$ \mathbf M\exp\{1/2\langle\mu\rangle_{\infty}-C\langle\mu\rangle_{\infty}^{1/2}\}<\infty $$
with any positive constant $C$ implies the uniform integrability of $Z_t$. In the case when $\mu_t$ is a stopped Wiener process the another condition (which is weaker and nonimproved in some sence) is given.