Unless I am mistaken, we know that an upper bound for the class number $h(D)$ of a real quadratic field $\mathbb{Q}(\sqrt{D})$ is $O(D^{1/2})$. Is the exponent of $1/2$ known to be the best possible?

Also, is there any better exponent known for the upper bound of $$\liminf_{D \rightarrow \infty} h(D) \, ?$$

Of course, the conjecture is that the exponent shoud be zero, but do we know any better than $1/2$?