Let
Ω
\Omega
be a Cartan domain of rank
r
r
and genus
p
p
and
B
ν
B_\nu
,
ν
>
p
−
1
\nu >p-1
, the Berezin transform on
Ω
\Omega
; the number
B
ν
f
(
z
)
B_{\nu }f(z)
can be interpreted as a certain invariant-mean-value of a function
f
f
around
z
z
. We show that a Lebesgue integrable function satisfying
f
=
B
ν
f
=
B
ν
+
1
f
=
⋯
=
B
ν
+
r
f
f=B_\nu f=B_{\nu +1}f=\dots =B_{\nu +r}f
,
ν
≥
p
\nu \ge p
, must be
M
\mathcal {M}
-harmonic. In a sense, this result is reminiscent of Delsarte’s two-radius mean-value theorem for ordinary harmonic functions on the complex
n
n
-space
C
n
\mathbf {C}^{n}
, but with the role of radius
r
r
played by the quantity
1
/
ν
1/\nu
.