For $\beta>1$, let $T_\beta$ be the $\beta$-transformation defined on $[0, 1)$. We study the sets of points whose orbits of $T_\beta$ have uniform Diophantine approximation properties. Precisely, for two given positive functions $\psi_1, \ \psi_2:{\Bbb N}\rightarrow{\Bbb R}^+$, define where $\gg$ means large enough. We calculate the Hausdorff dimension of the set ${\cal L}(\psi_1)\cap{\cal U}(\psi_2)$. As a corollary, we obtain the Hausdorff dimension of the set ${\cal U}(\psi_2)$. Our work generalizes the results of ^{[4]} where only exponential functions $\psi_1, \ \psi_2$ were taken into consideration.