Characterizations for inner functions in certain function spaces

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.