  For $\frac12<p<\infty$, $0<q<\infty$ and a certain two-sided doubling weight
$\omega$, we characterize those inner functions $\Theta$ for which
$$\|\Theta'\|_{A^{p,q}_\omega}^q=\int_0^1 \left(\int_0^{2\pi}
|\Theta'(re^{i\theta})|^p d\theta\right)^{q/p}
  \omega(r)\,dr<\infty.$$ Then we show a modified version of this result for
$p\ge q$. Moreover, two additional characterizations for inner functions whose
derivative belongs to the Bergman space $A_\omega^{p,p}$ are given.
