Let $K$ be a number field with Dirichlet zeta function $\zeta_K(s), $ we have the class number formula: $$\lim_{s \to 1^+} (s-1)\zeta_K(s) = \cfrac{2^{r_1}(2\pi)^{r_2}Rh}{m\sqrt{(|\Delta)|}}$$ where $r_1$ is the number of real embeddings of the field, $r_2$ for complex -nonconjugate- ones; $R$ for the regulator of $K, h$ is the class number of $K$, $m$ is the number of roots of unity in $K$ and $\Delta$ is the discriminant of $K$.
Using this formula, what can we say about $h$ and can we give a lower or upper bound for it?