On a Converse of Jensen's Discrete Inequality

Abstract

We give the best possible global bounds for a form of discrete Jensen's inequality. By some examples the fruitfulness of this result is shown.

1. Introduction

Throughout this paper represents a finite sequence of real numbers belonging to a fixed closed interval , and is a positive weight sequence associated with .

If is a convex function on , then the well-known Jensen's inequality [1, 2] asserts that

(1.1)

There are many important inequalities which are particular cases of Jensen's inequality among which are the weighted inequality, Cauchy's inequality, the Ky Fan and Hölder's inequalities.

One can see that the lower bound zero is of global nature since it does not depend on but only on and the interval whereupon is convex.

We give in [1] an upper global bound (i.e., depending on and only) which happens to be better than already existing ones. Namely, we prove that

(1.2)

with

(1.3)

Note that, for a (strictly) positive convex function , Jensen's inequality can also be stated in the form

(1.4)

It is not difficult to prove that is the best possible global lower bound for Jensen's inequality written in the above form. Our aim in this paper is to find the best possible global upper bound for (1.4). We will show with examples that by following this approach one may consequently obtain converses of some important inequalities.

2. Results

Our main result is contained in what follows.

Theorem 2.1.

Let be a (strictly) positive, twice continuously differentiable function on , and . One has that

(i)if is (strictly) convex function on , then

(2.1)

(ii)if is (strictly) concave function on , then

(2.2)

Both estimates are independent of .

The next assertion shows that (resp., ) exists and is unique.

Theorem 2.2.

There is unique such that

(2.3)

Of particular importance is the following theorem.

Theorem 2.3.

The expression represents the best possible global upper bound for Jensen's inequality written in the form (1.4).

3. Proofs

We will give proofs of the previous assertions related to the first part of Theorem 2.1. Proofs concerning concave functions go along the same lines.

Proof of Theorem 2.1.

We apply the method already shown in [1]. Namely, since , there is a sequence such that .

Hence,

(3.1)

Denoting , we get

(3.2)

Proof of Theorem 2.2.

For fixed , denote

(3.3)

We get with

(3.4)

Also,

(3.5)

Since is strictly convex on and , we conclude that is monotone decreasing on with . Since is continuous, there exists unique such that . Also , showing that is attained at the point . The proof is completed.

Proof of Theorem 2.3.

Let be an arbitrary global upper bound. By definition, the inequality

(3.6)

holds for arbitrary and .

In particular, for we obtain that as required.

4. Applications

In the sequel we will give some examples to demonstrate the fruitfulness of the assertions from Theorem 2.1. Since all bounds will be given as a combination of means from the Stolarsky class, here is its definition.

Stolarsky (or extended) two-parametric mean values are defined for positive values of as

(4.1)

means can be continuously extended on the domain

(4.2)

by the following:

(4.3)

and this form is introduced by Stolarsky in [3].

Most of the classical two variable means are special cases of the class . For example, is the arithmetic mean ,   is the geometric mean ,   is the logarithmic mean ,   is the identric mean , and so forth. More generally, the th power mean is equal to .

Example 4.1.

Taking , after an easy calculation it follows that . Therefore we consequently obtain the result.

Proposition 4.2.

If , then the inequality

(4.4)

holds for an arbitrary weight sequence .

This is the extended form of Schweitzer inequality.

Example 4.3.

For we get that the maximum of is attained at the point .

Hence, we have the following.

Proposition 4.4.

If , then the following means inequality

(4.5)

holds for an arbitrary weight sequence .

As a special case of the above inequality, that is, by putting and noting that imply , we obtain a converse of the well-known Cauchy's inequality.

Proposition 4.5.

If , then

(4.6)

In this form the Cauchy's inequality was stated in [2, page 80].

Note that the same result can be obtained from inequality (4.4) by taking .

Example 4.6.

Let . Since in this case is a concave function, applying the second part of Theorem 2.1, we get the following.

Proposition 4.7.

If , then

(4.7)

independently of .

In the limiting cases we obtain two important converses. Namely, writing (4.7) as

(4.8)

and, letting , the converse of generalized inequality arises.

Proposition 4.8.

If , then

(4.9)

Note that the right-hand side of (4.9) is exactly the Specht ratio (cf. [1]).

Analogously, writing (4.7) in the form

(4.10)

and taking the limit , one has the following.

Proposition 4.9.

If , then

(4.11)

Finally, putting in (4.7) , we obtain the converse of discrete Hölder's inequality.

Proposition 4.10.

If , then

(4.12)

It is interesting to compare (4.12) with the converse of Hölder's inequality for integral forms (cf. [4]).