Some New Hilbert's Type Inequalities

Abstract

Some new inequalities similar to Hilbert's type inequality involving series of nonnegative terms are established.

1. Introduction

In recent years, several authors [1–10] have given considerable attention to Hilbert's type inequalities and their various generalizations. In particular, in [1], Pachpatte proved somenew inequalities similar to Hilbert's inequality [11, page 226] involving series of nonnegative terms. The main purpose of this paper is to establish their general forms.

2. Main Results

In [1], Pachpatte established the following inequality involving series of nonnegative terms.

Theorem 2 A.

Let and let and be two nonnegative sequences of real numbers defined for and , where are natural numbers. Let and . Then

(2.1)

where

(2.2)

We first establish the following general form of inequality (2.1).

Theorem 2.1.

Let and . Let and be positive sequences of real numbers defined for and , where are natural numbers. Let and Then

(2.3)

where

(2.4)

Proof.

By using the following inequality (see [12]):

(2.5)

where is a constant, and , , we obtain

(2.6)

Similarly, we have

(2.7)

From (2.6) and (2.7), using Hölder's inequality [13] and the elementary inequality:

(2.8)

where , and we have

(2.9)

Dividing both sides of (2.9) by summing up over from 1 to first, then summing up over from 1 to , using again Hölder's inequality, then interchanging the order of summation, we obtain

(2.10)

This completes the proof.

Remark 2.2.

Taking , (2.3) becomes

(2.11)

Taking and changing , , and into , , and respectively, and with suitable changes, (2.11) reduces to Pachpatte [1, inequality (1)].

In [1], Pachpatte also established the following inequality involving series of nonnegative terms.

Theorem 2 B.

Let , be as defined in Theorem A. Let and be positive sequences for and where are natural numbers. Define and . Let and be real-valued, nonnegative, convex, submultiplicative functions defined on Then

(2.12)

where

(2.13)

Inequality (2.12) can also be generalized to the following general form.

Theorem 2.3.

Let , , and be as defined in Theorem 2.1. Let and be positive sequences for and . Define and Let and be real-valued, nonnegative, convex, submultiplicative functions defined on Then

(2.14)

where

(2.15)

Proof.

By the hypotheses, Jensen's inequality, and Hölder's inequality, we obtain

(2.16)

Similarly,

(2.17)

By (2.16) and (2.17), and using the elementary inequality:

(2.18)

where and we have

(2.19)

Dividing both sides of (2.19) by and summing up over from 1 to first, then summing up over from 1 to , using again inverse Hölder's inequality, and then interchanging the order of summation, we obtain

(2.20)

The proof is complete.

Remark 2.4.

Taking , (2.14) becomes

(2.21)

where

(2.22)

Taking and changing , , and into , , and respectively, and with suitable changes, (2.21) reduces to Pachpatte [1, Inequality ( 7)].

Theorem 2.5.

Let , , , , and , be as defined in Theorem 2.3. Define

(2.23)

for and where are natural numbers. Let and be real-valued, nonnegative, convex functions defined on Then

(2.24)

Proof.

By the hypotheses, Jensen's inequality, and Hölder's inequality, it is easy to observe that

(2.25)

(2.26)

Proceeding now much as in the proof of Theorems 2.1 and 2.3, and with suitable modifications, it is not hard to arrive at the desired inequality. The details are omitted here.

Remark 2.6.

In the special case where and , Theorem 2.5 reduces to the following result.

Theorem 2 C.

Let be as defined in Theorem B. Define and for and , where are natural numbers. Let and be real-valued, nonnegative, convex functions defined on Then

(2.27)

This is the new inequality of Pachpatte in [1,Theorem  4].

Remark 2.7.

Taking and in Theorem 2.5, and in view of , we obtain the following theorem.

Theorem 2 D.

Let , be as defined in Theorem A. Define and for and , where are natural numbers. Let and be real-valued, nonnegative, convex functions defined on Then

(2.28)

This is the new inequality of Pachpatte in [1,Theorem  3].