If
f
∈
C
[
−
1
,
1
]
f \in C[ - 1,1]
is real-valued, let
E
r
(
f
)
{E^{r}(f)}
and
E
c
(
f
)
{E^{c}(f)}
be the errors in best approximation to
f
f
in the supremum norm by rational functions of type
(
m
,
n
)
(m,n)
with real and complex coefficients, respectively. It has recently been observed that
E
c
(
f
)
>
E
r
(
f
)
{E^c}(f) > {E^r}(f)
can occur for any
n
⩾
1
n \geqslant 1
, but for no
n
⩾
1
n \geqslant 1
is it known whether
γ
m
n
=
inf
f
E
c
(
f
)
/
E
r
(
f
)
{\gamma _{mn}} = \inf _f\,{E^c}(f)/{E^{r}(f)}
is zero or strictly positive. Here we show that both are possible:
γ
01
>
0
{\gamma _{01}} > 0
, but
γ
m
n
=
0
{\gamma _{mn}} = 0
for
n
⩾
m
+
3
n \geqslant m + 3
. Related results are obtained for approximation on regions in the plane.