  The $\beta$-generalized quasi-geostrophic equation is studied in the range of
$\alpha \in (0, 1), \beta \in (1/2, 1), 1/2 < \alpha + \beta < 3/2$. When
$\alpha \in (1/2, 1), \beta \in (1/2, 1)$ such that $1 \leq \alpha + \beta <
3/2$, using the method introduced in [12] and [9], we prove global regularity
of the unique and analytic solution and when $\alpha \in (0, 1/2), \beta \in
(1/2, 1)$ such that $1/2 < \alpha + \beta < 1$, that there exists a constant
such that $\lVert
\nabla\theta_{0}\rVert_{L^{\infty}}^{2-2\alpha-2\beta}\lVert\theta_{0}\rVert_{L^{\infty}}^{2\alpha
+ 2\beta - 1} \leq c_{\alpha,\beta}$ implies global regularity.
