Suppose that
f
f
is in
L
2
(
Δ
)
{L^2}(\Delta )
where
Δ
\Delta
is the unit disk, and that
f
=
0
f = 0
outside
Δ
\Delta
. We show that then the Cauchy transform
C
f
\mathcal {C}\,f
of
f
f
, when restricted to
Δ
\Delta
, satisfies
|
|
C
f
|
|
2
≤
(
2
/
α
)
|
|
f
|
|
2
||\mathcal {C}\,f|{|_2} \leq (2/\alpha )||f|{|_2}
, where
α
≈
2.4048
\alpha \approx 2.4048
is the smallest positive zero of the Bessel function
J
0
{J_0}
. This inequality is sharp.