Resumen.
Sea $\Bbb D$ el
disco unitario en el plano complejo. Sea
$\varepsilon > 0$ y consideremos el sector
$\Sigma_{\varepsilon}
= \left\{z\in\Bbb C : |\arg z | < \varepsilon\right\}$.
Probaremos que para ciertas clases de funciones $f$ en el espacio
de Bergman con peso $A^p_{\alpha}\left(\Bbb D\right)$, que fijen
el origen, la norma se obtiene por integraci\'{o}n sobre
$f^{-1}(\Sigma_{\varepsilon})$, es decir, se cumple
$$ \int_{f^{-1}(\Sigma_{\varepsilon})}|f(z)|^pdA_{\alpha}(z) > \delta \|f\|^p_{\alpha,p}.$$
Este resultado extiende un teorema de Marshall y Smith
\cite{MS}.\par}
Abstract.
Let $\Bbb D$ be the open unit disk in the complex plane. For $\varepsilon > 0$ we consider the sector $\Sigma_{\varepsilon} = \{z\in\Bbb C : |\arg z | < \varepsilon \}$.
We will prove that for certain classes of functions $f$ in the
weighted Bergman's space $A_\alpha^p\left(\Bbb D\right)$ such that $f(0)=0$, the $A_\alpha^p$
norm
is obtained by integration over $f^{-1}(\Sigma_{\varepsilon})$, that is to
say
$$ \int_{f^{-1}(\Sigma_{\varepsilon})}|f(z)|^pdA_{\alpha}(z) > \delta \|f\|^p_{\alpha,p}.$$
This result extends a theorem of Marshall and Smith in
\cite{MS}.
pdAo""(z) > s'kfkpo"",p.
This result extends a theorem of Marshall and Smith in [MS].
