On Hilbert-Pachpatte Multiple Integral Inequalities

Abstract

We establish some multiple integral Hilbert-Pachpatte-type inequalities. As applications, we get some inverse forms of Pachpatte's inequalities which were established in 1998.

1. Introduction

In 1934, Hilbert [1] established the following well-known integral inequality.

If , , , and , then

(1.1)

where is the best value.

In recent years, considerable attention has been given to various extensions and improvements of the Hilbert inequality form different viewpoints [2–10]. In particular, Pachpatte [11] proved some inequalities similar to Hilbert's integral inequalities in 1998. In this paper, we establish some new multiple integral Hilbert-Pachpatte-type inequalities.

2. Main Results

Theorem 2.1.

Let , let , , where are positive real numbers, and define , for . Then for , and ,

(2.1)

Proof.

From the hypotheses and in view of inverse Hölder integral inequality (see [12]), it is easy to observe that

(2.2)

Let us note the following means inequality:

(2.3)

We obtain that

(2.4)

Integrating both sides of (2.4) over from to and using the special case of inverse Hölder integral inequality, we observe that

(2.5)

The proof is complete.

Remark 2.2.

Taking , to (2.1), (2.1) changes to

(2.6)

This is just an inverse inequality similar to the following inequality which was proved by Pachpatte [11]:

(2.7)

Theorem 2.3.

Let , , , and be as in Theorem 2.1. Let be n positive functions defined for , and define where are positive real numbers. Let be n real-valued nonnegative, concave, and super-multiplicative functions defined on . Then

(2.8)

where

(2.9)

Proof.

By using Jensen integral inequality (see [11]) and inverse Hölder integral inequality (see [12]) and noticing that are real-valued super-multiplicative functions, it is easy to observe that

(2.10)

In view of the means inequality and integrating two sides of (2.10) over from to and noticing Hölder integral inequality, we observe that

(2.11)

This completes the proof of Theorem 2.3.

Remark 2.4.

Taking , to (2.8), (2.8) changes to

(2.12)

where

(2.13)

This is just an inverse inequality similar to the following inequality which was proved by Pachpatte [11]:

(2.14)

where

(2.15)

Theorem 2.5.

Let , , , , and be as Theorem 2.3, and define for , where are positive real numbers. Let be real-valued, nonnegative, and concave functions on . Then

(2.16)

Proof.

From the hypotheses and by using Jensen integral inequality and the inverse Hölder integral inequality, we have

(2.17)

Hence

(2.18)

Remark 2.6.

Taking , to (2.16), (2.16) changes to

(2.19)

This is just an inverse inequality similar to the following inequality which was proved by Pachpatte [11]:

(2.20)

Remark 2.7.

In (2.20), if , then , . Therefore (2.20) changes to

(2.21)

This is just an inverse inequality similar to the following Inequality which was proved by Pachpatte [11]:

(2.22)