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 $$|Theta|_{A^{p,q}_omega}^q=int_0^1 left(int_0^{2pi} |Theta(re^{itheta})|^p dthetaright)^{q/p} omega(r),dr<infty.$$ Then we show a modified version of this result for $pge q$. Moreover, two additional characterizations for inner functions whose derivative belongs to the Bergman space $A_omega^{p,p}$ are given.