Let
K
(
z
,
z
)
,
R
(
z
,
z
)
K(z,z),R(z,z)
, and
C
(
z
)
C(z)
be the values of the Bergman kernel, the reduced Bergman kernel and the analytic capacity on an open Riemann surface
Ω
\Omega
(with respect to a local parameter z). Let
M
(
z
)
=
π
K
(
z
,
z
)
M(z) = \pi K(z,z)
and
S
(
z
)
=
π
R
(
z
,
z
)
S(z) = \sqrt {\pi R(z,z)}
. For
Ω
∉
O
G
\Omega \notin {O_G}
and for each integer
n
⩾
0
n \geqslant 0
, it is shown that
\[
C
(
n
+
1
)
(
n
+
2
)
⩽
(
n
+
1
)
!
(
∏
k
=
0
n
+
1
k
!
)
−
2
det
‖
M
j
k
¯
‖
j
,
k
=
0
n
,
{C^{(n + 1)(n + 2)}} \leqslant (n + 1)!{\left ( {\prod \limits _{k = 0}^{n + 1} {k!} } \right )^{ - 2}}\det \left \| {{M_{j\bar k}}} \right \|_{j,k = 0}^n,
\]
where
C
=
C
(
z
)
C = C(z)
and
M
j
k
¯
=
(
∂
j
+
k
/
∂
z
j
∂
z
¯
k
)
M
(
z
)
{M_{j\bar k}} = ({\partial ^{j + k}}/\partial {z^j}\partial {\bar z^k})M(z)
. Equality occurs if and only if
Ω
\Omega
is conformally equivalent to the unit disk less (possibly) a closed set of inner capacity zero. The special case of this result, namely when
n
=
0
n = 0
, is due to Hejhal and Suita. Let
κ
(
z
)
\kappa (z)
be the curvature of the “span metric”
S
(
z
)
|
d
z
|
S(z)|dz|
. As an attempt to resolve a conjecture of Suita, we also show that for
Ω
∉
O
A
D
,
κ
(
z
)
⩽
−
2
\Omega \notin {O_{AD}},\kappa (z) \leqslant - 2
for each
z
∈
Ω
z \in \Omega
. Both results are proved by studying suitable extremal problems.