The Best Lower Bound Depended on Two Fixed Variables for Jensen's Inequality with Ordered Variables

Abstract

We give the best lower bound for the weighted Jensen's discrete inequality with ordered variables applied to a convex function , in the case when the lower bound depends on , weights, and two given variables. Furthermore, under the same conditions, we give some sharp lower bounds for the weighted AM-GM inequality and AM-HM inequality.

1. Introduction

Let be a sequence of real numbers belonging to an interval , and let be a sequence of given positive weights associated to and satisfying . If is a convex function on , then the well-known discrete Jensen's inequality [1] states that

(1.1)

where

(1.2)

is the so-called Jensen's difference. The next refinement of Jensen's inequality was proven in [2], as a consequence of its Theorem 2.1, part (ii)

(1.3)

By (1.3), for fixed and , we get

(1.4)

In this paper, we will establish that the best lower bound of Jensen's difference for

(1.5)

has the expression

(1.6)

where

(1.7)

Logically, we need to have

(1.8)

Indeed, this inequality is equivalent to Jensen's inequality

(1.9)

2. Main Results

Theorem 2.1.

Let be a convex function on , and let such that

(2.1)

For fixed and (), Jensen's difference is minimal when

(2.2)

that is,

(2.3)

For proving Theorem 2.1, we will need the following three lemmas.

Lemma 2.2.

Let be nonnegative real numbers, and let be a convex function on . If such that and

(2.4)

then

(2.5)

Lemma 2.3.

Let be a convex function on , and let () such that

(2.6)

For fixed , where , Jensen's difference is minimal when

(2.7)

Lemma 2.4.

Let be a convex function on , and let () such that

(2.8)

For fixed , where , Jensen's difference is minimal when

(2.9)

Applying Theorem 2.1 for and using the substitutions , , we obtain

Corollary 2.5.

Let

(2.10)

and let be positive real numbers such that . Then,

(2.11)

with equality for

(2.12)

Using Corollary 2.5, we can prove the propositions below.

Proposition 2.6.

Let

(2.13)

and let be positive real numbers such that . If

(2.14)

then

(2.15)

with equality for . When , equality holds again for , , .

Proposition 2.7.

Let

(2.16)

and let be positive real numbers such that . Then,

(2.17)

with equality if and only if .

Remark 2.8.

For , from Proposition 2.6 we get the inequality

(2.18)

where

(2.19)

Equality in (2.18) holds for . If , then equality holds again for , , .

Remark 2.9.

For , from Proposition 2.7, we get the inequality

(2.20)

with equality if and only if .

Applying Theorem 2.1 for , we obtain

Corollary 2.10.

Let

(2.21)

and let be positive real numbers such that . Then,

(2.22)

with equality for

(2.23)

Remark 2.11.

For , from Corollary 2.10, we get the inequality

(2.24)

with equality for

(2.25)

If and , then (2.24) becomes

(2.26)

with equality for

(2.27)

In the case , from (2.26), we get

(2.28)

with equality for

(2.29)

Applying Theorem 2.1 for , we obtain the following.

Corollary 2.12.

Let

(2.30)

and let be positive real numbers such that . Then,

(2.31)

with equality for

(2.32)

Using Corollary 2.12, we can prove the following proposition.

Proposition 2.13.

Let

(2.33)

and let be positive real numbers such that . If

(2.34)

then

(2.35)

with equality for .

3. Proof of Lemmas

Proof of Lemma 2.2.

Since , there exist such that

(3.1)

In addition, from , we get

(3.2)

Applying Jensen's inequality twice, we obtain

(3.3)

and hence

(3.4)

Proof of Lemma 2.3.

We need to show that

(3.5)

Using Jensen's inequality

(3.6)

it suffices to prove that

(3.7)

which can be written as

(3.8)

where

(3.9)

Since and

(3.10)

by Lemma 2.2, the conclusion follows.

Proof of Lemma 2.4.

We need to prove that

(3.11)

By Jensen's inequality,we have

(3.12)

Therefore, it suffices to prove that

(3.13)

or, equivalently,

(3.14)

where

(3.15)

The inequality (3.14) follows from Lemma 2.2, since and

(3.16)

4. Proof of Theorem

Proof.

By Lemmas 2.3 and 2.4, it follows that for fixed , , Jensen's difference is minimal when and ; that is,

(4.1)

Therefore, towards proving (2.3), we only need to show that

(4.2)

Since

(4.3)

this inequality is a consequence of Jensen's inequality. Thus, the proof is completed.

5. Proof of Propositions

Proof of Proposition 2.6.

Using Corollary 2.5, we need to prove that

(5.1)

Since this inequality is homogeneous in and , and also in and , without loss of generality, assume that and . Using the notations and , where and , the inequality is equivalent to , where

(5.2)

with

(5.3)

We have

(5.4)

If , then

(5.5)

and if , then

(5.6)

Since for , and is increasing, , is increasing, and hence for . This concludes the proof.

Proof of Proposition 2.7.

Using Corollary 2.5, we need to prove that

(5.7)

Since this inequality is homogeneous in and , and also in and , we may set and . Using the notations and , where and , the inequality is equivalent to , where

(5.8)

We have

(5.9)

Since for , is strictly increasing, , and is strictly increasing, , is strictly increasing, and hence for .

Proof of Proposition 2.13.

Using Corollary 2.12, we need to prove that

(5.10)

This inequality is true if

(5.11)

For , we have

(5.12)

Also, for , we get

(5.13)

The proposition is proved.