The Gaussian product-inequality (GPI) conjecture is one of the most famous inequalities associated with Gaussian distributions and has attracted much attention. In this note, we investigate the quantitative versions of the two-dimensional Gaussian product inequalities. For any centered, nondegenerate, and two-dimensional Gaussian random vector $(X_{1}, X_{2})$
(
X
1
,
X
2
)
with $E[X_{1}^{2}]=E[X_{2}^{2}]=1$
E
[
X
1
2
]
=
E
[
X
2
2
]
=
1
and the correlation coefficient ρ, we prove that for any real numbers $\alpha _{1}, \alpha _{2}\in (-1,0)$
α
1
,
α
2
∈
(
−
1
,
0
)
or $\alpha _{1}, \alpha _{2}\in (0,\infty )$
α
1
,
α
2
∈
(
0
,
∞
)
, it holds that $$ {\mathbf{E}}\bigl[ \vert X_{1} \vert ^{\alpha _{1}} \vert X_{2} \vert ^{\alpha _{2}}\bigr]-{\mathbf{E}}\bigl[ \vert X_{1} \vert ^{ \alpha _{1}}\bigr]{\mathbf{E}}\bigl[ \vert X_{2} \vert ^{\alpha _{2}}\bigr]\ge f(\alpha _{1}, \alpha _{2}, \rho )\ge 0, $$
E
[
|
X
1
|
α
1
|
X
2
|
α
2
]
−
E
[
|
X
1
|
α
1
]
E
[
|
X
2
|
α
2
]
≥
f
(
α
1
,
α
2
,
ρ
)
≥
0
,
where the function $f(\alpha _{1}, \alpha _{2}, \rho )$
f
(
α
1
,
α
2
,
ρ
)
will be given explicitly by the Gamma function and is positive when $\rho \neq 0$
ρ
≠
0
. When $-1<\alpha _{1}<0$
−
1
<
α
1
<
0
and $\alpha _{2}>0$
α
2
>
0
, Russell and Sun (Statist. Probab. Lett. 191:109656, 2022) proved the “opposite Gaussian product inequality”, of which we will also give a quantitative version. These quantitative inequalities are derived by employing the hypergeometric functions and the generalized hypergeometric functions.