Derivatives of inner functions in weighted Bergman spaces and the Schwarz-Pick lemma

We characterize those radial doubling weights $\omega$ for which the Schwarz-Pick lemma applied to the derivative of any inner function in the norm of the Bergman space $A^p_\omega$ does not cause any essential loss of information. The approach we employ is based on operator theory and leads to a characterization of when the linear average operator \[ T(f)(z)=\frac {\int _{|z|}^1f\left (s\frac {z}{|z|}\right ) ds}{1-|z|} \] is bounded from $A^p_\omega$ to $L^p_\omega$. The characterizing integral condition is self-improving and therefore $T:A^p_\omega \to L^p_\omega$ is bounded if and only if $T:A^{p-\varepsilon }_\omega \to L^{p-\varepsilon }_\omega$ is bounded for all sufficiently small $\varepsilon >0$. This study also reveals the fact that, under appropriate hypothesis on $\omega$, the average operator $T:A^p_\omega \to L^p_\omega$ is bounded if and only if the Bergman projection $P:L^p_\omega \to L^p_\omega$ is bounded if and only if the classical Hilbert operator $\mathcal {H}:L^{p+1}_{\widehat {\omega }}([0,1))\to A^{p+1}_\omega$ is bounded, where $\widehat {\omega }(r)=\int _r^1\omega (s) ds$.