  For $0<p<\infty$, we give a complete description of nonnegative radial weight
functions $\omega$ on the open unit disk $\mathbb{D}$ such that $$
  \int_{\mathbb{D}} |f'(z)|^p (1-|z|^2)^{p-2}\omega(z)dA(z)<\infty
  $$ if and only if $$
\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^p}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}(1-|z|^2)^{\tau}(1-|\zeta|^2)^{\sigma}\omega(\zeta)dA(z)A(\zeta)<\infty
$$ for all analytic functions $f$ in $\mathbb{D}$, where $\tau$ and $\sigma$
are some real numbers. As applications, we give some geometric descriptions of
functions in Besove type spaces $B_p(\omega)$ with doubling weights, and
characterize the boundedness and compactness of Hankel type operators related
to Besov type spaces with radial B\'ekoll\'e-Bonami weights. Some special cases
of our results are new even for some standard weighted Besov spaces.
