Let G denote an arbitrary bounded regular region in the plane and
H
2
(
G
)
{H_2}\left ( G \right )
the analytic Hardy class on G with index 2. We show that the generalized isoperimetric inequality
1
π
∬
G
|
φ
(
z
)
ψ
(
z
)
|
2
d
x
d
y
⩽
1
2
π
∫
∂
G
|
φ
(
z
)
|
2
|
d
z
|
1
2
π
∫
∂
G
|
ψ
(
z
)
|
2
|
d
z
|
(
z
=
x
+
i
y
)
\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
φ
\varphi
and
ψ
∈
H
2
(
G
)
\psi \, \in \,{H_2}\left ( G \right )
. We also determine necessary and sufficient conditions for equality.