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More About HermiteHadamard Inequalities, Cauchy's Means, and Superquadracity
Journal of Inequalities and Applications volume 2010, Article number: 102467 (2010)
Abstract
New results associated with HermiteHadamard inequalities for superquadratic functions are given. A set of Cauchy's type means is derived from these HermiteHadamardtype inequalities, and its logconvexity and monotonicity are proved.
1. Introduction
The following inequality:
is holding for any convex function, that is, well known in the literature as the HermiteHadamard inequality (see [1, page 137]). In many areas of analysis applications of HermiteHadamard 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
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
Definition(see [ 9 , Definition]). A function is exponentially convex if it is continuous and
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
for every and every ,
(iii) is continuous and
for every
If is exponentially convex function, then is a logconvex function:
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
then is continuous function.
The next two sections are about mean value theorems, positive semidefiniteness, exponential convexity, logconvexity, Cauchy means, and their monotonicity, that are associated with HermiteHadamard inequalities for superquadratic functions.
2. Mean Value Theorems
Definition B. Let be an integrable function; for one defines a linear functional as
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
Consider the functions defined as
Then and are increasing functions. If also then they are superquadratic functions.
Theorem 2.3.
If and , then the following equality holds:
Proof.
Suppose that is bounded, that is, and . Using instead of in (1.3) we get
Similarly, using instead of in (1.3) we get
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
provided the denominators are not equal to zero. If K is invertible then
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:
When goes to 2, we have
where
When goes to 2 we have
where is defined above and
In when goes to , we have
where
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
where denotes and denotes . In limiting case when goes to 2s is equal to
where denotes and denotes . When goes to we have,
where
When goes to in , we have
where
Definition C. Let be an integrable function, for One defines a linear functional as
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,
Proof.
Suppose that is bounded, that is, and . Using from Lemma 2.2 instead of in (1.4), we get
Similarly, using from Lemma 2.2 instead of in (1.4) we get
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
provided the denominators are not equal to zero. If T is invertible, then
is a new mean.
If we put and in (2.27) we have new mean defined as follows.
Definition. We define for , , as follows:
where denotes and denotes . In the limiting case we have which is equal to
where denotes
where
In when goes to , we have
where
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:
where denotes and denotes . In limiting case we have which is equal to
where
When approaches to ,
where
where
3. Positive Semidefiniteness, Exponential Convexity, and LogConvexity
Lemma 3.1 (see [12, Lemma ]).
Consider the function for defined as
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,
(b)One has
that is, is logconvex in the Jensen sense.
(c)The function is exponentially convex.
(d) is logconvex, that is, for where one has
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
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 logconvex and we get (3.4).
Corollary 3.3.
One has the following
(i)For ,
(ii)For ,
(iii)For ,
(iv)For ,
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 positivesemidefinite matrix, that is,
(b)One has
that is, is logconvex in the Jensen sense.
(c)The function is exponentially convex.
(d) is logconvex, that is, for where one has
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 logconvex function, and if , then the following inequality is valid,
Theorem 3.6.
For such that and , one has for as in Definition
Proof.
According to Theorem 3.2, defined above is logconvex; so Lemma 3.5 implies that for such that and we have
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
Proof.
As defined above is logconvex, Lemma 3.5 implies that for such that and we have
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
Proof.
According to Theorem 3.4, defined above is logconvex; so Lemma 3.5 implies that for such that and we have
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
Proof.
As defined above is logconvex, Lemma 3.5 implies that for such that and we have
By substituting , and , such that , , we get the result, and for , we can consider the limiting case.
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Acknowledgment
This research was partially funded by Higher Education Commission, Pakistan. The research of the third author was supported by the Croatian Ministry of Science, Education and Sports under the Research grant no. 11711708890888.
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Abramovich, S., Farid, G. & Pečarić, J. More About HermiteHadamard Inequalities, Cauchy's Means, and Superquadracity. J Inequal Appl 2010, 102467 (2010). https://doi.org/10.1155/2010/102467
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DOI: https://doi.org/10.1155/2010/102467