Quantitative versions of the two-dimensional Gaussian product inequalities

Abstract

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 $(X_{1}, X_{2})$ with $E[X_{1}^{2}]=E[X_{2}^{2}]=1$ and the correlation coefficient ρ, we prove that for any real numbers $\alpha _{1}, \alpha _{2}\in (-1,0)$ or $\alpha _{1}, \alpha _{2}\in (0,\infty )$, it holds that

$$ {\mathbf{E}}\bigl[ \vert X_{1} \vert ^{\alpha _{1}} \vert X_{2} \vert ^{\alpha _{2}}\bigr]-{\mathbf{E}}\bigl[ \vert X_{1} \vert ^{ \alpha _{1}}\bigr]{\mathbf{E}}\bigl[ \vert X_{2} \vert ^{\alpha _{2}}\bigr]\ge f(\alpha _{1}, \alpha _{2}, \rho )\ge 0, $$

where the function $f(\alpha _{1}, \alpha _{2}, \rho )$ will be given explicitly by the Gamma function and is positive when $\rho \neq 0$. When $-1<\alpha _{1}<0$ and $\alpha _{2}>0$, 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.

1 Introduction

The Gaussian product-inequality (GPI) conjecture is one of the most famous inequalities associated with Gaussian distributions, which states that for any d-dimensional, real-valued, and centered Gaussian random vector $(X_{1}, \ldots , X_{d})$,

$$\begin{aligned}& {\mathbf{E}} \Biggl[\prod_{j=1}^{d}X_{j}^{2m} \Biggr]\ge \prod_{j=1}^{d}{ \mathbf{E}} \bigl[X_{j}^{2m} \bigr],\quad m\in \mathbb{N}. \end{aligned}$$

(1.1)

In [12], Li and Wei proposed the following improved version of the GPI conjecture:

$$\begin{aligned}& {\mathbf{E}} \Biggl[\prod_{j=1}^{d} \vert X_{j} \vert ^{\alpha _{j}} \Biggr]\ge \prod _{j=1}^{d}{ \mathbf{E}} \bigl[ \vert X_{j} \vert ^{\alpha _{j}} \bigr], \end{aligned}$$

(1.2)

where $\alpha _{j},j=1, 2,\ldots , d$, are nonnegative real numbers.

Up to now, the GPI (1.1) and the GPI (1.2) are still open, however, some special cases have been proved by using various tools. Frenkel [6] proved (1.1) with $m=1$ (or (1.2) for the case $\alpha _{j}=2$) by using algebraic methods. Malicet et al. [13] gave an analytic proof to Frenkel’s result among other issues, such as giving new lower bounds on homogeneous polynomials (see [13, Theorem 1.5]), and presenting a new result supporting the U-conjecture (see [13, Theorem 1.7], the U-conjecture is still open). Wei [21] proved a stronger version of (1.2) for $\alpha _{j}\in (-1,0)$ as follows:

$$\begin{aligned} {\mathbf{E}} \Biggl[\prod_{j=1}^{d} \vert X_{j} \vert ^{\alpha _{j}} \Biggr]\ge {\mathbf{E}} \Biggl[\prod_{j=1}^{k} \vert X_{j} \vert ^{\alpha _{j}} \Biggr]{\mathbf{E}} \Biggl[ \prod _{j=k+1}^{d} \vert X_{j} \vert ^{\alpha _{j}} \Biggr],\quad \forall 1\leq k \leq d-1. \end{aligned}$$

(1.3)

From Karlin and Rinott [10], we know that (1.1) and (1.2) hold for $\mathbf{X}=(X_{1},\ldots ,X_{d})$ when the density of $|{\mathbf{X}}|=(|X_{1}|,|X_{2}|,\ldots , |X_{d}|)$ satisfies the so-called condition $\mathbf{MTP} _{2}$, and for any nondegenerate, 2-dimensional, and centered Gaussian random vector $(X_{1}, X_{2})$, $(|X_{1}|,|X_{2}|)$ has an $\mathbf{MTP} _{2}$ density. Thus, (1.1) and (1.2) hold for $d=2$.

Lan et al. [11] used the hypergeometric functions to prove the following 3-dimensional GPI: for any $m_{1}, m_{2}\in \mathbb{N}$ and any centered Gaussian random vector $(X_{1}, X_{2}, X_{3})$,

$$\begin{aligned} {\mathbf{E}} \bigl[X_{1}^{2m_{1}}X_{2}^{2m_{2}}X_{3}^{2m_{2}} \bigr]\ge { \mathbf{E}} \bigl[X_{1}^{2m_{1}} \bigr]{ \mathbf{E}} \bigl[X_{2}^{2m_{2}} \bigr]{\mathbf{E}} \bigl[X_{3}^{2m_{2}} \bigr]. \end{aligned}$$

(1.4)

Russell and Sun [20] proved the following 3-dimensional GPI: for any $m_{2}, m_{3}\in \mathbb{N}$ and any centered Gaussian random vector $(X_{1}, X_{2}, X_{3})$,

$$\begin{aligned} {\mathbf{E}} \bigl[X_{1}^{2}X_{2}^{2m_{2}}X_{3}^{2m_{3}} \bigr]\ge {\mathbf{E}} \bigl[X_{1}^{2} \bigr]{ \mathbf{E}} \bigl[X_{2}^{2m_{2}} \bigr]{\mathbf{E}} \bigl[X_{3}^{2m_{3}} \bigr]. \end{aligned}$$

(1.5)

Herry et al. [9] proved the following 3-dimensional GPI: for any $m_{1},m_{2}, m_{3}\in \mathbb{N}$ and any centered Gaussian random vector $(X_{1}, X_{2}, X_{3})$,

$$\begin{aligned} {\mathbf{E}} \bigl[X_{1}^{2m_{1}}X_{2}^{2m_{2}}X_{3}^{2m_{3}} \bigr]\ge { \mathbf{E}} \bigl[X_{1}^{2m_{1}} \bigr]{ \mathbf{E}} \bigl[X_{2}^{2m_{2}} \bigr]{\mathbf{E}} \bigl[X_{3}^{2m_{3}} \bigr]. \end{aligned}$$

(1.6)

Genest and Ouimet [7] proved that if there exists a matrix $C\in [0, +\infty )^{d\times d}$ such that $(X_{1}, X_{2}, \ldots , X_{d})=(Z_{1}, Z_{2}, \ldots , Z_{d})C$ in law, where $(Z_{1}, Z_{2}, \ldots , Z_{d})$ is a d-dimensional, standard Gaussian random vector, the following stronger version of (1.1) holds:

$$ {\mathbf{E}} \Biggl[\prod_{j=1}^{d}X_{j}^{2m_{j}} \Biggr]\ge \mathbf{E} \Biggl[\prod_{j=1}^{k}X_{j}^{2m_{j}} \Biggr]\mathbf{E} \Biggl[\prod_{j=k+1}^{n}X_{j}^{2m_{j}} \Biggr], \quad m_{j}\in \mathbb{N},j=1,\ldots ,n, \forall 1\leq k\leq n-1. $$

(1.7)

Russell and Sun [17] proved among other things that (1.7) holds if all the correlation coefficients are nonnegative. Edelmann et al. [5] extended (1.7) to the multivarite gamma distributions. As to other related work, we refer to Genest and Ouiment [8], Russell and Sun [18], and Russell and Sun [19].

Recently, De et al. [3] obtained a range of quantitative correlation inequalities, including the quantitative version of the Gaussian correlation inequality proved by Royen [16], and the quantitative version of the well-known Fortuin–Kasteleyn–Ginibre (FKG) inequality for monotone functions over any finite product probability space. A further related work is De et al. [4].

To our understanding, quantitative correlation inequalities contain more information than qualitative correlation inequalities, and can imply the latter. A quantitative inequality with respect to (w.r.t. for short) the corresponding qualitative inequality is like the law of iterated logarithm w.r.t. the strong law of large numbers. In this note, we consider the quantitative versions of the two-dimensional Gaussian product inequalities.

For any centered, nondegenerate, and two-dimensional Gaussian random vector $(X_{1}, X_{2})$ with the correlation coefficient ρ, without loss of generality, we assume that ${\mathbf{E}}[X_{1}^{2}]=\mathbf{E}[X_{2}^{2}]=1$. We will prove that for any real numbers $\alpha _{1}, \alpha _{2}\in (-1,0)$ or $\alpha _{1}, \alpha _{2}\in (0,\infty )$, it holds that

$$ {\mathbf{E}}\bigl[ \vert X_{1} \vert ^{\alpha _{1}} \vert X_{2} \vert ^{\alpha _{2}}\bigr]-{\mathbf{E}}\bigl[ \vert X_{1} \vert ^{ \alpha _{1}}\bigr]{\mathbf{E}}\bigl[ \vert X_{2} \vert ^{\alpha _{2}}\bigr]\ge f(\alpha _{1}, \alpha _{2}, \rho )\ge 0, $$

where the function $f(\alpha _{1}, \alpha _{2}, \rho )$ will be given explicitly by the Gamma function and is positive when $\rho \neq 0$. When $-1<\alpha _{1}<0$ and $\alpha _{2}>0$, Russell and Sun [18] 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.

The rest of this note is organized as follows. In Sect. 2, we present the main results and some corollaries. In Sect. 3, we give the proofs. In Sect. 4, we give some remarks. In Sect. 5, we give the conclusion and introduce some related questions.

2 Main results

Throughout this note, any Gaussian random variable is assumed to be real-valued, nondegenerate, and standard. Our main results are as follows.

Theorem 2.1

Let $(X_{1},X_{2})$ be centered, bivariate Gaussian random variables with the correlation coefficient ρ satifying $|\rho |<1$. Then, for any real numbers $\alpha _{1}, \alpha _{2}\in (-1,0)$ or $\alpha _{1}, \alpha _{2}\in (0,\infty )$,

$$\begin{aligned} \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}{ \vert X_{2} \vert }^{ \alpha _{2}}\bigr]- \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}\bigr] \textbf{E}\bigl[{ \vert X_{2} \vert }^{\alpha _{2}}\bigr]\geq f( \alpha _{1}, \alpha _{2}, \rho ), \end{aligned}$$

(2.1)

where the nonnegative function $f(\cdot )$ is defined by

$$\begin{aligned}& f(\alpha _{1}, \alpha _{2}, \rho ) \\& \quad :=\textstyle\begin{cases} \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}\alpha _{1}\alpha _{2}\rho ^{2}}{2\pi} \Gamma (\frac{\alpha _{1}+1}{2})\Gamma (\frac{\alpha _{2}+1}{2}),& \textit{if}\ -1< \alpha _{1},\alpha _{2}< 0\ \textit{or}\ 0< \alpha _{1}, \alpha _{2}\leq 2\ \textit{or}\ \alpha _{1}, \alpha _{2}>2, \\ \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}\alpha _{1}\alpha _{2}\rho ^{2}}{4\sqrt{\pi}} \Gamma (\frac{\alpha _{1}+\alpha _{2}-1}{2}), &\textit{if}\ \alpha _{1}>2,0< \alpha _{2}\le 2\ \textit{or}\ 0< \alpha _{1}\le 2, \alpha _{2}>2. \end{cases}\displaystyle \end{aligned}$$

When the real numbers $\alpha _{1}$ and $\alpha _{2}$ in Theorem 2.1 have opposite signs, Russell and Sun [18] proved the following opposite GPI:

$$\begin{aligned} \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}{ \vert X_{2} \vert }^{ \alpha _{2}}\bigr]\leq \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}\bigr] \textbf{E}\bigl[{ \vert X_{2} \vert }^{\alpha _{2}}\bigr]. \end{aligned}$$

(2.2)

We have the quantitative version of the GPI (2.2) as follows.

Theorem 2.2

Let $(X_{1},X_{2})$ be centered, bivariate Gaussian random variables with the correlation coefficient ρ satifying $|\rho |<1$. Let $\alpha _{1}\in (-1,0)$ and $\alpha _{2}\in (0,\infty )$.

• (i) When $0<\alpha _{2}\le 2$,

  $$\begin{aligned}& \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}\alpha _{1}\alpha _{2}\rho ^{2}}{2\pi} \Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr) \Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr)F \biggl(1- \frac{\alpha _{1}}{2},1- \frac{\alpha _{2}}{2}; \frac{3}{2};1 \biggr) \\& \quad \leq \textbf{E} \bigl[{ \vert X_{1} \vert }^{\alpha _{1}}{ \vert X_{2} \vert }^{\alpha _{2}} \bigr]-\textbf{E} \bigl[{ \vert X_{1} \vert }^{ \alpha _{1}} \bigr]\textbf{E} \bigl[{ \vert X_{2} \vert }^{\alpha _{2}} \bigr] \\& \quad \leq \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}\alpha _{1}\alpha _{2}\rho ^{2}}{2\pi} \Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr) \Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr) \leq 0, \end{aligned}$$

  (2.3)

  where $F(\cdot )$ is the hypergeometric function defined by

  $$\begin{aligned} F(a,b;c;z):=\sum_{n=0}^{+\infty} \frac{(a)_{n}(b)_{n}}{(c)_{n}}\cdot \frac{z^{n}}{n!},\ \vert z \vert \leq 1, \end{aligned}$$

  (2.4)

  and, for $\alpha \in \mathbb{R}$,

  $$\begin{aligned} (\alpha )_{n}:=\textstyle\begin{cases} \alpha (\alpha +1)\dots (\alpha +n-1), & n \geq 1, \\ 1, & n=0,\alpha \neq 0. \end{cases}\displaystyle \end{aligned}$$

• (ii) When $\alpha _{2}>2$,

  $$\begin{aligned}& \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}\alpha _{1}\alpha _{2}\rho ^{2}}{2\pi} \Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr) \Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr) \\& \quad \leq \textbf{E} \bigl[{ \vert X_{1} \vert }^{\alpha _{1}}{ \vert X_{2} \vert }^{\alpha _{2}} \bigr]-\textbf{E} \bigl[{ \vert X_{1} \vert }^{ \alpha _{1}} \bigr]\textbf{E} \bigl[{ \vert X_{2} \vert }^{\alpha _{2}} \bigr] \\& \quad \leq \min \biggl\{ \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}\alpha _{1}\alpha _{2} \rho ^{2}}{2\pi}\Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr)\Gamma \biggl(\frac{\alpha _{2}+1}{2} \biggr) F \biggl(1- \frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2};\frac{3}{2};1 \biggr), 0 \biggr\} . \end{aligned}$$

  (2.5)

By Theorem 2.1, we have the following corollaries.

Corollary 2.3

Let $\alpha _{1},\alpha _{2}\in \{1,2\}$ in Theorem 2.1, then we have

$$\begin{aligned}& \textbf{E}\bigl[{ \vert X_{1} \vert } { \vert X_{2} \vert }\bigr]-\textbf{E}\bigl[{ \vert X_{1} \vert }\bigr]\textbf{E}\bigl[{ \vert X_{2} \vert }\bigr]\geq \frac{\rho ^{2}}{\pi}, \\& \textbf{E}\bigl[{ \vert X_{1} \vert } { X_{2}}^{2} \bigr]-\textbf{E}\bigl[{ \vert X_{1} \vert }\bigr]\textbf{E} \bigl[{X_{2}}^{2}\bigr]\geq \frac{\sqrt{2}\rho ^{2}}{\sqrt{\pi}}, \\& \textbf{E}\bigl[{ X_{1} }^{2}{X_{2}}^{2} \bigr]-\textbf{E}\bigl[{X_{1}}^{2}\bigr] \textbf{E} \bigl[{ X_{2} }^{2}\bigr]\geq 2\rho ^{2}. \end{aligned}$$

Corollary 2.4

Under the conditions of Theorem 2.1, suppose that $\alpha _{2}=1$, and $\alpha _{1}$ is an integer satisfying $\alpha _{1}=m>2$. Then,

$$\begin{aligned} \textbf{E}\bigl[{ \vert X_{1} \vert }^{m}{ \vert X_{2} \vert }\bigr]-\textbf{E}\bigl[{ \vert X_{1} \vert }^{m}\bigr]\textbf{E}\bigl[{ \vert X_{2} \vert }\bigr]\geq \textstyle\begin{cases} \frac{(m-2)!! m\rho ^{2}}{\sqrt{2\pi}}, & \textit{if } m \textit{ is even}, \\ \frac{(m-2)!! m\rho ^{2}}{2},& \textit{if } m\textit{ is odd}. \end{cases}\displaystyle \end{aligned}$$

Corollary 2.5

Under the conditions of Theorem 2.1, suppose that $\alpha _{1}$ and $\alpha _{2}$ are integers and $\alpha _{1}=m$, $\alpha _{2}=n$ with $m,n\in (2,\infty )$,

• (i) if m, n are both even integers, then

  $$\begin{aligned} \textbf{E}\bigl[{ X_{1} }^{m}{X_{2}}^{n} \bigr]-\textbf{E}\bigl[{X_{1}}^{m}\bigr]\textbf{E} \bigl[{ X_{2} }^{n}\bigr]&\geq \frac{(m-1)!!(n-1)!!mn\rho ^{2}}{2}; \end{aligned}$$

• (ii) if m, n are both odd integers, then

  $$\begin{aligned} \textbf{E}\bigl[{ \vert X_{1} \vert }^{m}{ \vert X_{2} \vert }^{n}\bigr]- \textbf{E}\bigl[{ \vert X_{1} \vert }^{m}\bigr]\textbf{E}\bigl[{ \vert X_{2} \vert }^{n}\bigr]& \geq \frac{(m-1)!!(n-1)!!mn\rho ^{2}}{\pi}; \end{aligned}$$

• (iii) if one of m and n is odd and the other is even, then

  $$\begin{aligned} \textbf{E}\bigl[{ \vert X_{1} \vert }^{m}{ \vert X_{2} \vert }^{n}\bigr]- \textbf{E}\bigl[{ \vert X_{1} \vert }^{m}\bigr]\textbf{E}\bigl[{ \vert X_{2} \vert }^{n}\bigr]& \geq \frac{(m-1)!!(n-1)!!mn\rho ^{2}}{\sqrt{2\pi}}. \end{aligned}$$

3 Proofs

In this section, we will give the proofs of Theorems 2.1 and 2.2. First, we recall some properties of hypergeometric functions and generalized hypergeometric functions in Sect. 3.1. The proofs of Theorems 2.1 and 2.2 will be presented in Sects. 3.2 and 3.3, respectively.

3.1 Preliminaries

Definition 3.1

(see Andrews et al. [1, Chap. 2] or Rainville [15, Chap. 5])

The generalized hypergeometric function $_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z)$ is defined by

$$\begin{aligned}& _{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z):=\sum _{k=0}^{+ \infty} \frac{(a_{1})_{n} \dots (a_{p})_{n}}{(b_{1})_{n}\dots (b_{q})_{n}}\cdot \frac{z^{n}}{n!}, \end{aligned}$$

(3.1)

where

$$\begin{aligned} (\alpha )_{n}:=\textstyle\begin{cases} \alpha (\alpha +1)\dots (\alpha +n-1), & n \geq 1, \\ 1, & n=0,\alpha \neq 0. \end{cases}\displaystyle \end{aligned}$$

In particular, when $p=2$ and $q=1$, $_{2}F_{1}(a_{1},a_{2};b_{1};z)$ is called the hypergeometric function that is written as $F(a,b;c;z)$ in short (see (2.4)).

Theorem 3.2

(see Andrews et al. [1, Theorem 2.2.5] or Rainville [15, Theorem 21])

If $|z|<1$,

$$\begin{aligned}& F(a,b;c;z)=(1-z)^{c-a-b}F(c-a,c-b;c;z). \end{aligned}$$

(3.2)

Theorem 3.3

(see Rainville [15, Theorem 28])

If $p\le q+1$, Re($b_{1}$)> Re($a_{1}$)>0, no one of $b_{1}, b_{2}, \ldots , b_{q}$ is zero or a negative integer, and $|z|<1$,

$$\begin{aligned}& _{p}F_{q}(a_{1}, a_{2},\ldots , a_{p}; b_{1}, b_{2}, \ldots , b_{q};z) \\& \quad =\frac{\Gamma (b_{1})}{\Gamma (a_{1})\Gamma (b_{1}-a_{1})} \int ^{1}_{0} t^{(a_{1}-1)}(1-t)^{(b_{1}-a_{1}-1)}_{p-1}F_{q-1}( a_{2},\ldots , a_{p}; b_{2}, \ldots , b_{q};z). \end{aligned}$$

(3.3)

If $p\le q$, the condition $|z|<1$ may be omitted.

Theorem 3.4

(see Rainville [15, Theorem 18])

If $\operatorname{Re} (c-a-b)>0$ and if c is neither zero nor a negative integer,

$$\begin{aligned}& F(a,b;c;1)=\frac{\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}. \end{aligned}$$

(3.4)

In addition, from Andrews et al. [1, (2.5.1)], we know that the hypergeometric function satisfies the following differential equation:

$$\begin{aligned}& \frac{d}{dz}F(a,b;c;z)=\frac{ab}{c}F(a+1,b+1;c+1;z). \end{aligned}$$

(3.5)

3.2 Proof of Theorem 2.1

Obviously, we can assume that $\rho \neq 0$. From the density function of a centered Gaussian random variable and the definition of the Gamma function, we have that for $i=1,2$,

$$\begin{aligned} \textbf{E}\bigl[{ \vert X_{i} \vert }^{\alpha _{i}}\bigr] =& \int _{-\infty}^{+ \infty}\frac{1}{\sqrt{2\pi}}{ \vert x \vert }^{\alpha _{i}}e^{- \frac{x^{2}}{2}}\,dx=\frac{\sqrt{2}}{\sqrt{\pi}} \int _{0}^{+\infty}x^{ \alpha _{i}}e^{-\frac{x^{2}}{2}}\,dx \\ =&\frac{2^{\frac{\alpha _{i}}{2}}}{\sqrt{\pi}} \int _{0}^{+\infty}y^{ \frac{\alpha _{i}-1}{2}}e^{-y}\,dy= \frac{2^{\frac{\alpha _{i}}{2}}}{\sqrt{\pi}} \Gamma \biggl( \frac{\alpha _{i}+1}{2} \biggr). \end{aligned}$$

(3.6)

Then,

$$\begin{aligned}& \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}\bigr]\textbf{E}\bigl[{ \vert X_{2} \vert }^{\alpha _{2}}\bigr]= \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}} }{\pi} \Gamma \biggl( \frac{\alpha _{1}+1}{2} \biggr)\Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr). \end{aligned}$$

(3.7)

Since $|\rho |<1$, by Nabeya [14], we know that

$$\begin{aligned}& \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}{ \vert X_{2} \vert }^{ \alpha _{2}}\bigr]= \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}} }{\pi} \Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr)\Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr) F \biggl(-\frac{\alpha _{1}}{2},- \frac{\alpha _{2}}{2};\frac{1}{2};\rho ^{2} \biggr), \end{aligned}$$

(3.8)

where $F(\cdot )$ is the hypergeometric function defined by (2.4).

It follows from (3.7) and (3.8) that

$$\begin{aligned}& \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}{ \vert X_{2} \vert }^{ \alpha _{2}}\bigr]- \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}\bigr] \textbf{E}\bigl[{ \vert X_{2} \vert }^{\alpha _{2}}\bigr] \\& \quad =\frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}}{\pi} \Gamma \biggl( \frac{\alpha _{1}+1}{2} \biggr)\Gamma \biggl(\frac{\alpha _{2}+1}{2} \biggr) \biggl[F \biggl(-\frac{\alpha _{1}}{2},- \frac{\alpha _{2}}{2}; \frac{1}{2};\rho ^{2} \biggr)-1 \biggr]. \end{aligned}$$

(3.9)

Since $\frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}}{\pi} \Gamma ( \frac{\alpha _{1}+1}{2})\Gamma (\frac{\alpha _{2}+1}{2})>0$ for any $-1<\alpha _{1}, \alpha _{2}<0$ or $\alpha _{1}, \alpha _{2}>0$, it is enough to find the lower bound of $F(-\frac{\alpha _{1}}{2},-\frac{\alpha _{2}}{2};\frac{1}{2};\rho ^{2})-1$. First, we have that

$$\begin{aligned} F \biggl(-\frac{\alpha _{1}}{2},-\frac{\alpha _{2}}{2}; \frac{1}{2}; \rho ^{2} \biggr)-1 &=\sum _{k=1}^{+\infty} \frac{(-\frac{\alpha _{1}}{2})_{k}(-\frac{\alpha _{2}}{2})_{k}}{ (\frac{1}{2})_{k}}\cdot \frac{({\rho ^{2}})^{k}}{k!} \\ &=\sum_{k=0}^{+\infty} \frac{(-\frac{\alpha _{1}}{2})_{k+1} (-\frac{\alpha _{2}}{2})_{k+1}}{(\frac{1}{2})_{k+1}} \cdot \frac{({\rho ^{2}})^{k+1}}{(k+1)!} \\ &=\frac{\rho ^{2}\alpha _{1}\alpha _{2}}{2}\sum_{k=0}^{+ \infty} \frac{(1-\frac{\alpha _{1}}{2})_{k}(1-\frac{\alpha _{2}}{2})_{k}(1)_{k}}{ (\frac{3}{2})_{k}(2)_{k}}\cdot \frac{{\rho ^{2}}^{k}}{(k)!} \\ &=\frac{\rho ^{2}\alpha _{1}\alpha _{2}}{2}\ {_{3}F_{2}} \biggl(1- \frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2},1;\frac{3}{2},2;\rho ^{2} \biggr), \end{aligned}$$

(3.10)

where $_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z)$ is the generalized hypergeometric function defined by (3.1). By the property of the generalized hypergeometric function in (3.3) and (3.10), we obtain that

$$\begin{aligned} F \biggl(-\frac{\alpha _{1}}{2},-\frac{\alpha _{2}}{2}; \frac{1}{2}; \rho ^{2} \biggr)-1 &=\frac{\rho ^{2}\alpha _{1}\alpha _{2}}{2} \frac{\Gamma (2)}{\Gamma (1)\Gamma (1)} \int _{0}^{1} F \biggl(1- \frac{\alpha _{1}}{2},1- \frac{\alpha _{2}}{2};\frac{3}{2};\rho ^{2}t \biggr)\,dt \\ &=\frac{\alpha _{1}\alpha _{2}}{2} \int _{0}^{\rho ^{2}} F \biggl(1- \frac{\alpha _{1}}{2},1- \frac{\alpha _{2}}{2};\frac{3}{2};h \biggr)\,dh. \end{aligned}$$

(3.11)

Secondly, by (3.5) and the Euler transformation (3.2), we have that

$$\begin{aligned}& \frac{d}{dh}F \biggl(1-\frac{\alpha _{1}}{2},1- \frac{\alpha _{2}}{2}; \frac{3}{2};h \biggr) \\& \quad=\frac{2}{3} \biggl(1-\frac{\alpha _{1}}{2} \biggr) \biggl(1- \frac{\alpha _{2}}{2} \biggr) F \biggl(2-\frac{\alpha _{1}}{2},2- \frac{\alpha _{2}}{2};\frac{5}{2};h \biggr) \\& \quad=\frac{2}{3} \biggl(1-\frac{\alpha _{1}}{2} \biggr) \biggl(1- \frac{\alpha _{2}}{2} \biggr) (1-h)^{ \frac{1+\alpha _{1}+\alpha _{2}}{2}-2} F \biggl( \frac{1}{2}+ \frac{\alpha _{1}}{2},\frac{1}{2}+ \frac{\alpha _{2}}{2};\frac{5}{2};h \biggr). \end{aligned}$$

(3.12)

Therefore, in order to estimate $F(-\frac{\alpha _{1}}{2},-\frac{\alpha _{2}}{2};\frac{1}{2};\rho ^{2})-1$, we divide it into the following two cases.

Case I: When $-1<\alpha _{1},\alpha _{2}<0$ or $0<\alpha _{1},\alpha _{2}\leq 2$, or $\alpha _{1}, \alpha _{2}>2$.

In this case, $(1-\frac{\alpha _{1}}{2}) (1-\frac{\alpha _{2}}{2} )\geq 0$ and $\alpha _{1}\alpha _{2}>0$. Thus, the derivative of $\frac{d}{dh}F(1-\frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2}; \frac{3}{2};h)$ in (3.12) is nonnegative. Therefore, by (3.11) we obtain that

$$\begin{aligned} F \biggl(-\frac{\alpha _{1}}{2},-\frac{\alpha _{2}}{2};\frac{1}{2}; \rho ^{2} \biggr)-1& \geq \frac{\alpha _{1}\alpha _{2}}{2} \int _{0}^{ \rho ^{2}} F \biggl(1-\frac{\alpha _{1}}{2},1- \frac{\alpha _{2}}{2}; \frac{3}{2};0 \biggr)\,dh =\frac{\alpha _{1}\alpha _{2}}{2} \cdot \rho ^{2}. \end{aligned}$$

Thus, by the above statements, we obtain that

$$\begin{aligned} \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}{ \vert X_{2} \vert }^{ \alpha _{2}}\bigr]-\textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}\bigr] \textbf{E}\bigl[{ \vert X_{2} \vert }^{\alpha _{2}}\bigr]&\geq \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}} }{\pi} \Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr)\Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr) \frac{\alpha _{1}\alpha _{2}}{2} \cdot \rho ^{2} \\ &= \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}\alpha _{1}\alpha _{2}\rho ^{2}}{2\pi} \Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr) \Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr). \end{aligned}$$

Case II: When $\alpha _{1}> 2$ and $0<\alpha _{2}\le 2$ or $0<\alpha _{1}\le 2$ and $\alpha _{2}>2$.

Without loss of generality, it is assumed that $\alpha _{1}>2$ and $0<\alpha _{2}\le 2$. Then, $(1-\frac{\alpha _{1}}{2})(1-\frac{\alpha _{2}}{2})\le 0$ and $\alpha _{1}\alpha _{2}>0$. Thus, in this case, by (3.12), we know that

$$\begin{aligned} \frac{d}{dh}F \biggl(1-\frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2}; \frac{3}{2};h \biggr)\le 0. \end{aligned}$$

Then, $F(1-\frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2};\frac{3}{2};h)$ reaches its minimum at $h=1$, and the minimum value is

$$\begin{aligned} F \biggl(1-\frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2};\frac{3}{2};1 \biggr) = \frac{\Gamma (\frac{3}{2})\Gamma (\frac{\alpha _{1}+\alpha _{2}}{2}-\frac{1}{2})}{\Gamma (\frac{\alpha _{1}}{2}+\frac{1}{2})\Gamma (\frac{\alpha _{2}}{2}+\frac{1}{2})}, \end{aligned}$$

where the equality holds by (3.4). Then, from (3.11), we obtain

$$\begin{aligned} F \biggl(-\frac{\alpha _{1}}{2},-\frac{\alpha _{2}}{2};\frac{1}{2}; \rho ^{2} \biggr)-1 &\geq \frac{\alpha _{1}\alpha _{2}}{2}\cdot \rho ^{2} \frac{\Gamma (\frac{3}{2})\Gamma (\frac{\alpha _{1}+\alpha _{2}}{2}-\frac{1}{2})}{\Gamma (\frac{\alpha _{1}}{2}+\frac{1}{2})\Gamma (\frac{\alpha _{2}}{2}+\frac{1}{2})}. \end{aligned}$$

From (3.9), we obtain

$$\begin{aligned} & \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}{ \vert X_{2} \vert }^{ \alpha _{2}}\bigr]-\textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}\bigr] \textbf{E}\bigl[{ \vert X_{2} \vert }^{\alpha _{2}}\bigr] \\ &\quad\geq \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}}{\pi} \Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr) \Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr)\frac{\alpha _{1}\alpha _{2}}{2}\cdot \rho ^{2} \frac{\Gamma (\frac{3}{2})\Gamma (\frac{\alpha _{1}+\alpha _{2}}{2}-\frac{1}{2})}{ \Gamma (\frac{\alpha _{1}}{2}+\frac{1}{2})\Gamma (\frac{\alpha _{2}}{2}+\frac{1}{2})} \\ &\quad= \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}\alpha _{1}\alpha _{2} \rho ^{2}}{4\sqrt{\pi}} \Gamma \biggl( \frac{\alpha _{1}+\alpha _{2}-1}{2} \biggr). \end{aligned}$$

The proof is complete.

3.3 Proof of Theorem 2.2

From the proof of Theorem 2.1, we know that

$$\begin{aligned} & \textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}{ \vert X_{2} \vert }^{ \alpha _{2}}\bigr]-\textbf{E}\bigl[{ \vert X_{1} \vert }^{\alpha _{1}}\bigr] \textbf{E}\bigl[{ \vert X_{2} \vert }^{\alpha _{2}}\bigr] \\ &\quad=\frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}} }{\pi} \Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr) \Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr) \biggl[F \biggl(-\frac{\alpha _{1}}{2},- \frac{\alpha _{2}}{2};\frac{1}{2};\rho ^{2} \biggr)-1 \biggr], \end{aligned}$$

(3.13)

$$\begin{aligned} &F \biggl(-\frac{\alpha _{1}}{2},-\frac{\alpha _{2}}{2};\frac{1}{2}; \rho ^{2} \biggr)-1 =\frac{\alpha _{1}\alpha _{2}}{2} \int _{0}^{\rho ^{2}} F \biggl(1-\frac{\alpha _{1}}{2},1- \frac{\alpha _{2}}{2};\frac{3}{2};h \biggr)\,dh, \end{aligned}$$

(3.14)

and

$$\begin{aligned} &\frac{d}{dh}F \biggl(1-\frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2}; \frac{3}{2};h \biggr) \\ &\quad =\frac{2}{3} \biggl(1-\frac{\alpha _{1}}{2} \biggr) \biggl(1- \frac{\alpha _{2}}{2} \biggr) (1-h)^{ \frac{1+\alpha _{1}+\alpha _{2}}{2}-2} F \biggl( \frac{1}{2}+ \frac{\alpha _{1}}{2},\frac{1}{2}+ \frac{\alpha _{2}}{2};\frac{5}{2};h \biggr). \end{aligned}$$

Case I: $0<\alpha _{2}\le 2$. In this case $1-\frac{\alpha _{2}}{2}\geq 0$ and thus

$$\begin{aligned} \frac{d}{dh}F \biggl(1-\frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2}; \frac{3}{2};h \biggr)&\geq 0. \end{aligned}$$

Then, from (3.14), we obtain

$$\begin{aligned} \frac{\alpha _{1}\alpha _{2}\rho ^{2}}{2} F \biggl(1- \frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2}; \frac{3}{2};1 \biggr) \leq F \biggl(-\frac{\alpha _{1}}{2},- \frac{\alpha _{2}}{2}; \frac{1}{2};\rho ^{2} \biggr)-1 \leq \frac{\alpha _{1}\alpha _{2}\rho ^{2}}{2}\le 0, \end{aligned}$$

which together with (3.13) implies (2.3).

Case II: $\alpha _{2}>2$. In this case, $1-\frac{\alpha _{2}}{2}<0$, and thus

$$\begin{aligned} \frac{d}{dh}F \biggl(1-\frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2}; \frac{3}{2};h \biggr)&\leq 0. \end{aligned}$$

Then, from (3.14), it can be derived that

$$\begin{aligned} \frac{\alpha _{1}\alpha _{2}\rho ^{2}}{2}\leq F \biggl(- \frac{\alpha _{1}}{2},- \frac{\alpha _{2}}{2};\frac{1}{2};\rho ^{2} \biggr)-1 \leq \frac{\alpha _{1}\alpha _{2}\rho ^{2}}{2} F \biggl(1- \frac{\alpha _{1}}{2},1-\frac{\alpha _{2}}{2}; \frac{3}{2};1 \biggr), \end{aligned}$$

which together with (3.13) implies (2.5).

The proof is complete. □

4 Remarks

Remark 4.1

If ρ in Theorem 2.1 satisfies that $|\rho |=1$, then $X_{2}=\pm X_{1}$ in law and thus $\mathbf{E}[|X_{1}|^{\alpha _{1}}|X_{2}|^{\alpha _{2}}]=\mathbf{E}[|X_{1}|^{ \alpha _{1}+\alpha _{2}}]$. From (3.6) and $\Gamma (\frac{1}{2})=\sqrt{\pi}$, we know that

$$\begin{aligned}& \mathbf{E}\bigl[ \vert X_{1} \vert ^{\alpha _{1}} \vert X_{2} \vert ^{\alpha _{2}}\bigr]-\mathbf{E}\bigl[ \vert X_{1} \vert ^{ \alpha _{1}} \vert \bigr] \mathbf{E}\bigl[ \vert X_{2} \vert ^{\alpha _{2}}\bigr] \\& \quad =\mathbf{E}\bigl[ \vert X_{1} \vert ^{\alpha _{1}+\alpha _{2}} \bigr]-\mathbf{E}\bigl[ \vert X_{1} \vert ^{ \alpha _{1}} \vert \bigr] \mathbf{E}\bigl[ \vert X_{2} \vert ^{\alpha _{2}} \bigr] \\& \quad =\frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}}{\sqrt{\pi}} \Gamma \biggl(\frac{\alpha _{1}+\alpha _{2}+1}{2} \biggr)- \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}} }{\pi} \Gamma \biggl(\frac{\alpha _{1}+1}{2} \biggr)\Gamma \biggl( \frac{\alpha _{2}+1}{2} \biggr) \\& \quad =\frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}}{\pi}\Gamma \biggl( \frac{\alpha _{1}+1}{2} \biggr) \Gamma \biggl(\frac{\alpha _{2}+1}{2} \biggr) \biggl[ \frac{\Gamma (\frac{\alpha _{1}+\alpha _{2}+1}{2})\Gamma (\frac{1}{2})}{\Gamma (\frac{\alpha _{1}+1}{2}) \Gamma (\frac{\alpha _{2}+1}{2})} -1 \biggr] \\& \quad = \frac{2^{\frac{\alpha _{1}+\alpha _{2}}{2}}}{\pi} \Gamma \biggl( \frac{\alpha _{1}+1}{2} \biggr)\Gamma \biggl(\frac{\alpha _{2}+1}{2} \biggr) \biggl[ F \biggl(-\frac{\alpha _{1}}{2}, - \frac{\alpha _{2}}{2};\frac{1}{2};1 \biggr)-1 \biggr], \end{aligned}$$

the last equality follows from (3.4). That is to say, (3.9) still holds in this case. Hence, the result of Theorem 2.1 still holds if $|\rho |=1$ and one of the following two conditions holds:

• (i) $\alpha _{1},\alpha _{2}\in (0,\infty )$;

• (ii) $\alpha _{1},\alpha _{2}\in (-1,0)$ and $\alpha _{1}+\alpha _{2}\in (-1,0)$.

Remark 4.2

If ρ in Theorem 2.2 satisfies that $|\rho |=1$, then by Remark 4.1 and the proof of Theorem 2.2 we know that the result of Theorem 2.2 still holds in this case.

5 Conclusions

In this note, we gave several quantitative versions of the two-dimensional Gaussian product inequalities, which can imply the corresponding qualitative inequalities and contain more information. The main results are Theorems 2.1 and 2.2.

We hope that this note is a good starting point in this aspect and can stimulate more related work. In fact, we can ask the following questions:

Question 1

How about the quantitative versions of the 3-dimensional Gaussian product inequalities (1.4), (1.5), and (1.6)?

Question 2

How about the quantitative version of the product inequality (1.7)?

Question 3

How about the quantitative version of the product inequality (1.3)?