Let
δ
\delta
be a positive real constant and let
G
G
be an abelian group (written additively) in which division by 2 is uniquely performable. Every unbounded complex-valued function
f
f
on
G
G
satisfying the inequality
\[
|
f
(
x
+
y
)
f
(
x
−
y
)
−
f
(
x
)
2
+
f
(
y
)
2
|
⩽
δ
for all
x
,
y
∈
G
\left | {f(x + y)f(x - y) - f{{(x)}^2} + f{{(y)}^2}} \right | \leqslant \delta \quad {\text {for all }}x,y \in G
\]
has to be a solution of the sine functional equation
\[
f
(
x
+
y
)
f
(
x
−
y
)
=
f
(
x
)
2
−
f
(
y
)
2
for all
x
,
y
∈
G
.
f(x + y)f(x - y) = f{(x)^2} - f{(y)^2}\quad {\text {for all }}x,y \in G.
\]