Characterizations for inner functions in certain function spaces

For $\frac{1}{2}<p<\mathrm{\infty }$, $0<q<\mathrm{\infty }$ and a certain two-sided doubling weight $\omega$, we characterize those inner functions $\mathrm{\Theta }$ for which $$\mathrm{\parallel }{\mathrm{\Theta }}^{\mathrm{\prime }}{\mathrm{\parallel }}_{A_{\omega }^{p,q}}^q={\int }_0^1{({\int }_0^{2\pi }\mathrm{\mid }{\mathrm{\Theta }}^{\mathrm{\prime }}(r{e}^{i\theta }){\mathrm{\mid }}^{p}d\theta )}^{q\mathrm{/}p}\omega (r)dr<\mathrm{\infty }\mathrm{.}$$ 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.