  We establish the well/ill-posedness theories for the inviscid
$\alpha$-surface quasi-geostrophic ($\alpha$-SQG) equations in H\"older spaces,
where $\alpha = 0$ and $\alpha = 1$ correspond to the two-dimensional Euler
equation in the vorticity formulation and SQG equation of geophysical
significance, respectively. We first prove the local-in-time well-posedness of
$\alpha$-SQG equations in $C([0,T);C^{0,\beta}(\mathbb{R}^2))$ with $\beta \in
(\alpha,1)$ for some $T>0$. We then analyze the strong ill-posedness in
$C^{0,\alpha}(\mathbb{R}^2)$ constructing smooth solutions to the $\alpha$-SQG
equations that exhibit $C^{0,\alpha}$--norm growth in a short time. In
particular, we develop the nonexistence theory for $\alpha$-SQG equations in
$C^{0,\alpha}(\mathbb{R}^2)$.
