The Bergman norm and the Szegő norm

Abstract:

Let G denote an arbitrary bounded regular region in the plane and ${H_2}\left ( G \right )$ the analytic Hardy class on G with index 2. We show that the generalized isoperimetric inequality \begin{multline} \frac {1}{\pi } \iint \limits _G {{{\left | {\varphi \left ( z \right )\psi \left ( z \right )} \right |}^{2 }}dx dy \leqslant } \frac {1}{{2\pi }} \int _{\partial G}{{{\left | \varphi (z) \right |}^{2}}}\left | dz \right | \frac {1}{2\pi } \int _{\partial G}{{{\left | \psi (z) \right |}^{2}} \left | dz \right |} (z = x + iy) \end{multline} holds for any $\varphi$ and $\psi \in {H_2}\left ( G \right )$. We also determine necessary and sufficient conditions for equality.