Let
A
A
denote a complex unital Banach algebra with Hermitian elements
(
A
)
(A)
. We show that if
F
F
is an analytic function from a connected open set
D
D
into
A
A
such that
F
(
z
)
F(z)
is normal
(
F
(
z
)
=
u
(
z
)
+
i
υ
(
z
)
(F(z) = u(z) + i\upsilon (z)
, where
u
(
z
)
u(z)
,
υ
(
z
)
∈
H
(
A
)
\upsilon (z) \in H(A)
and
u
(
z
)
υ
(
z
)
=
υ
(
z
)
u
(
z
)
)
u(z)\upsilon (z) = \upsilon (z)u(z))
for each
z
∈
D
z \in D
, then
F
(
z
)
F
(
w
)
=
F
(
w
)
F
(
z
)
F(z)F(w) = F(w)F(z)
for all
w
w
,
z
∈
D
z \in D
. This generalizes a theorem of Globevnik and Vidav concerning operator-valued analytic functions. As a corollary, it follows that an essentially normal-valued analytic function has an essentially commutative range.