We prove the following subadditive property of the error function:
\[
erf
(
x
)
=
2
π
∫
0
x
e
−
t
2
d
t
(
x
∈
R
)
.
\mbox {erf}\,(x)=\frac {2}{\sqrt {\pi }}\int _0^x e^{-t^2}dt \quad {(x\in \mathbf {R})}.
\]
Let
a
a
and
b
b
be real numbers. The inequality
\[
erf
(
(
x
+
y
)
a
)
b
>
erf
(
x
a
)
b
+
erf
(
y
a
)
b
\mbox {erf}\,\bigl ((x+y)^a\bigr )^b> \mbox {erf}\,(x^a)^b + \mbox {erf}\,(y^a)^b
\]
holds for all positive real numbers
x
x
and
y
y
if and only if
a
b
≤
1
ab\leq 1
.