If
δ
>
0
\delta > 0
, G is an abelian group and f is a complex-valued function defined on G such that
|
f
(
x
+
y
)
+
f
(
x
−
y
)
−
2
f
(
x
)
f
(
y
)
|
⩽
δ
|f(x + y) + f(x - y) - 2f(x)f(y)| \leqslant \delta
for all
x
,
y
∈
G
x,y \in G
, then either
|
f
(
x
)
|
⩽
(
1
+
1
+
2
δ
)
/
2
|f(x)| \leqslant (1 + \sqrt {1 + 2\delta } )/2
for all
x
∈
G
x \in G
or
f
(
x
+
y
)
+
f
(
x
−
y
)
=
2
f
(
x
)
f
(
y
)
f(x + y) + f(x - y) = 2f(x)f(y)
for all
x
,
y
∈
G
x,y \in G
.