More About Hermite-Hadamard Inequalities, Cauchy's Means, and Superquadracity

Abstract

New results associated with Hermite-Hadamard inequalities for superquadratic functions are given. A set of Cauchy's type means is derived from these Hermite-Hadamard-type inequalities, and its log-convexity and monotonicity are proved.

1. Introduction

The following inequality:

(1.1)

is holding for any convex function, that is, well known in the literature as the Hermite-Hadamard inequality (see [1, page 137]). In many areas of analysis applications of Hermite-Hadamard inequality appear for different classes of functions with and without weights; see for convex functions, for example, [2, 3]. Also some useful mappings are defined connected to this inequality see in [4–6]. Here we focus on a class of functions which are superquadratic and analogs and refinements of (1.1) are applied to obtain results useful in analysis.

Now we present definitions, theorems, and results that we use in this paper.

The following definition is given in [7].

Definition A. A function is superquadratic provided that for all there exists a constant such that

(1.2)

for all . One says that is subquadratic if is a superquadratic function.

The followings theorem is given in [8] and is used in our main results:

Theorem 1.1.

Let be an integrable superquadratic function; then for one has

(1.3)

(1.4)

Definition(see [ 9 , Definition]). A function is exponentially convex if it is continuous and

(1.5)

for all and all choices and such that

Proposition 1.2 (see [9, Proposition ]).

Let . The following are equivalent:

(i) is exponentially convex,

(ii) is continuous and

(1.6)

for every and every ,

(iii) is continuous and

(1.7)

for every

Corollary 1.3 (see [8, 9]).

If is exponentially convex function, then is a log-convex function:

(1.8)

for all .

Remark 1.4.

In Definition and Proposition 1.2 it is sufficient to require measurability and finiteness almost every where in place of continuity because of the following theorem (see [10, page 105, Theorem .1b]and [11]): if the function is measurable and finite almost everywhere and if in addition

(1.9)

then is continuous function.

The next two sections are about mean value theorems, positive semidefiniteness, exponential convexity, log-convexity, Cauchy means, and their monotonicity, that are associated with Hermite-Hadamard inequalities for superquadratic functions.

2. Mean Value Theorems

Definition B. Let be an integrable function; for one defines a linear functional as

(2.1)

It is clear from (1.3) Theorem 1.1 of that; if is superquadratic function; then .

In [7] we have the following Lemma.

Lemma 2.1.

Suppose that is continuously differentiable and . If is superadditive or is increasing, then is superquadratic.

Lemma 2.2 (see [12, Lemma ]).

Let such that

(2.2)

Consider the functions defined as

(2.3)

Then and are increasing functions. If also then they are superquadratic functions.

Theorem 2.3.

If and , then the following equality holds:

(2.4)

Proof.

Suppose that is bounded, that is, and . Using instead of in (1.3) we get

(2.5)

Similarly, using instead of in (1.3) we get

(2.6)

By combining the above two inequalities we get that there exists such that (2.4) holds. Moreover if (for example) is bounded from above we have that (2.5) is valid. Also (2.4) holds when is not bounded.

We omit the proofs of Theorems 2.4 and 2.6 as they are similar to the proofs in [9, 13–16].

Theorem 2.4.

If , , and , then one has

(2.7)

provided the denominators are not equal to zero. If K is invertible then

(2.8)

is a new mean.

It is easy to check that the set of functions , satisfies Lemma 2.1. Therefore if we put and in (2.8), we have a new mean defined as follows.

Definition. One defines new mean for , and , , as follows:

(2.9)

When goes to 2, we have

(2.10)

where

(2.11)

When goes to 2 we have

(2.12)

where is defined above and

(2.13)

In when goes to , we have

(2.14)

where

(2.15)

If we put and in (2.8), then by the substitution, , we have a new mean defined as

Definition. Let and , one defines Cauchy mean as

(2.16)

where denotes and denotes . In limiting case when goes to 2s is equal to

(2.17)

where denotes and denotes . When goes to we have,

(2.18)

where

(2.19)

When goes to in , we have

(2.20)

where

(2.21)

Definition C. Let be an integrable function, for One defines a linear functional as

(2.22)

It is clear from (1.4) Theorem 1.1 of that if is superquadratic function, then .

Theorem 2.5.

If and , then the following equality holds,

(2.23)

Proof.

Suppose that is bounded, that is, and . Using from Lemma 2.2 instead of in (1.4), we get

(2.24)

Similarly, using from Lemma 2.2 instead of in (1.4) we get

(2.25)

By combining the above two inequalities we get that there exist such that (2.23) holds. Moreover if (for example) is bounded from above we have that (2.24) is valid. Also (2.23) holds when is not bounded.

Theorem 2.6.

If , and then, one has

(2.26)

provided the denominators are not equal to zero. If T is invertible, then

(2.27)

is a new mean.

If we put and in (2.27) we have new mean defined as follows.

Definition. We define for , , as follows:

(2.28)

where denotes and denotes . In the limiting case we have which is equal to

(2.29)

where denotes

(2.30)

where

(2.31)

In when goes to , we have

(2.32)

where

(2.33)

If we put and in (2.27), then by the substitution we have new mean defined as follows.

Definition. Let , and , , one defines Cauchy mean as follows:

(2.34)

where denotes and denotes . In limiting case we have which is equal to

(2.35)

where

(2.36)

When approaches to ,

(2.37)

where

(2.38)

where

(2.39)

3. Positive Semidefiniteness, Exponential Convexity, and Log-Convexity

Lemma 3.1 (see [12, Lemma ]).

Consider the function for defined as

(3.1)

Then, with the convention , is superquadratic.

Theorem 3.2.

For defined in (2.1) one has the following.

(a)The matrix , , is a positive semidefinite matrix, that is,

(3.2)

(b)One has

(3.3)

that is, is log-convex in the Jensen sense.

(c)The function is exponentially convex.

(d) is log-convex, that is, for where one has

(3.4)

Proof.

• (a) Define the function , where Then,

  

  (3.5)

and This implies that is superquadratic, so using this in the place of in (2.1) we have

(3.6)

From this we have that the matrix is positive semidefinite.

• (b) It is a simple consequence of for

• (c) Since we have , so is continuous for all ; then by (3.6) and Proposition 1.2 we have that is exponentially convex.

• (d) As is continuous then we have that is log-convex and we get (3.4).

Corollary 3.3.

One has the following

(i)For ,

(3.7)

(ii)For ,

(3.8)

(iii)For ,

(3.9)

(iv)For ,

(3.10)

Proof.

By applying Theorem 3.2(b) with and respectively, we get the result.

Similar to Theorem 3.2 we get the following.

Theorem 3.4.

For defined in (2.22) one has the following.

(a)The matrix , , is a positive-semidefinite matrix, that is,

(3.11)

(b)One has

(3.12)

that is, is log-convex in the Jensen sense.

(c)The function is exponentially convex.

(d) is log-convex, that is, for where one has

(3.13)

Proof.

The proof is the same as the proof of Theorem 3.2.

In the next results we use the continuity of and .

When log f is convex we see that (also see [13])

Lemma 3.5.

Let be log-convex function, and if , then the following inequality is valid,

(3.14)

Theorem 3.6.

For such that and , one has for as in Definition

(3.15)

Proof.

According to Theorem 3.2, defined above is log-convex; so Lemma 3.5 implies that for such that and we have

(3.16)

From the continuity of we get our result for , , and for we can consider limiting case.

Theorem 3.7.

For such that and , one has for as in Definition

(3.17)

Proof.

As defined above is log-convex, Lemma 3.5 implies that for such that and we have

(3.18)

By substituting , and , such that , , we get the result, and for , we can consider the limiting case.

Theorem 3.8.

For such that and , one has

(3.19)

Proof.

According to Theorem 3.4, defined above is log-convex; so Lemma 3.5 implies that for such that and we have

(3.20)

From the continuity of we get our result for , ; and for we can consider limiting case.

Theorem 3.9.

For such that and , one has

(3.21)

Proof.

As defined above is log-convex, Lemma 3.5 implies that for such that and we have

(3.22)

By substituting , and , such that , , we get the result, and for , we can consider the limiting case.