Integral Inequalities of Hermite-Hadamard Type for Functions Whose 3rd Derivatives Are s-Convex ()
1. Introduction
The following definition is well known in the literature.
Definition 1.1. A function
is said to be convex if
![](https://www.scirp.org/html/15-7401087\c6f09270-699f-4f73-b674-cdd1de544481.jpg)
holds for all
and
.
In [1,2], among others, the concepts of so-called quasiconvex and s-convex functions in the second sense was introduced as follows.
Definition 1.2 ([1]). A function
is said to be quasi-convex if
![](https://www.scirp.org/html/15-7401087\54dd1672-bd74-428f-a712-d4290eb238bd.jpg)
holds for all
and
.
Definition 1.3 ([2]). Let
A function
is said to be s-convex in the second sense if
![](https://www.scirp.org/html/15-7401087\229f5d67-2d11-497b-8ee2-1fc1813e0494.jpg)
for all
and
.
If
is a convex function on
with
and
, Then we have Hermite-Hardamard’s inequality
. (1.1)
Hermite-Hadamard inequality (1.1) has been refined or generalized for convex, s-convex, and quasi-convex functions by a number of mathematicians. Some of them can be reformulated as follows.
Theorem 1.1 ([3, Theorems 2.2 and 2.3]). Let
be a differentiable mapping on
,
with
.
(1) If
is convex on
, then
. (1.2)
(2) If the new mapping
is convex on
for
, then
![](https://www.scirp.org/html/15-7401087\171c339e-2358-45eb-8b5d-eed4a5c5f619.jpg)
Theorem 1.2 ([4, Theorems 1 and 2]). Let
be a differentiable function on
and
with
, and let
. If
is convex on
, then
(1.3)
and
(1.4)
Theorem 1.3 ([5, Theorems 2.3 and 2.4]). Let
be differentiable on
,
with
, and let
. If
is convex on
, then
![](https://www.scirp.org/html/15-7401087\cc636c12-0090-4e94-8b51-633844af3654.jpg)
and
(1.5)
Theorem 1.4 ([6, Theorems 1 and 3]). Let
be differentiable on
and
with
.
(1) If
is s-convex on
for some fixed
and
, then
(1.6)
(2) If
is s-convex on
for some fixed
and
, then
(1.7)
Theorem 1.5 ([7, Theorem 2]). Let
be an absolutely continuous function on
such that
for
with
. If
is quasi-convex on
, then
![](https://www.scirp.org/html/15-7401087\8ffb04a0-aa03-42cc-816d-b18aca4bfa4a.jpg)
In recent years, some other kinds of Hermite-Hadamard type inequalities were created in, for example, [8-17], especially the monographs [18,19], and related references therein.
In this paper, we will find some new inequalities of Hermite-Hadamard type for functions whose third derivatives are s-convex and apply these inequalities to discover inequalities for special means.
2. A Lemma
For finding some new inequalities of Hermite-Hadamard type for functions whose third derivatives are
-convex, we need a simple lemma below.
Lemma 2.1. Let
be a three times differentiable function on
with
and
. If
, then
(2.1)
Proof. By integrating by part, we have
![](https://www.scirp.org/html/15-7401087\8cebd609-ca76-40f7-a04d-0f807e3a670f.jpg)
The proof of Lemma 2.1 is complete.
3. Some New Hermite-Hadamard Type Inequalities
We now utilize Lemma 2.1, Hölder’s inequality, and others to find some new inequalities of Hermite-Hadamard type for functions whose third derivatives are s-convex.
Theorem 3.1. Let
be a three times differentiable function on
such that
for
with
. If
is s-convex on
for some fixed
and
, then
(3.1)
Proof. Since
is s-convex on
, by Lemma 2.1 and Hölder’s inequality, we have
![](https://www.scirp.org/html/15-7401087\9c311620-3939-4ef9-ba88-ee97109f5fa4.jpg)
where
![](https://www.scirp.org/html/15-7401087\68e71217-1923-4fed-a645-87183526ed6d.jpg)
and
![](https://www.scirp.org/html/15-7401087\76eda0cc-dd97-45e6-a6f4-b66f5a15be07.jpg)
Thus, we have
![](https://www.scirp.org/html/15-7401087\10d6ec44-be68-44cc-9b92-187b54688839.jpg)
The proof of Theorem 3.1 is complete.
Corollary 3.1.1. Under conditions of Theorem 3.11) if
, then
(3.2)
2) if
, then
![](https://www.scirp.org/html/15-7401087\d3e55d67-764e-4fbc-aee6-375f9aecdae9.jpg)
Theorem 3.2. Let
be a three times differentiable function on
such that
for
with
. If
is s-convex on
for some fixed
and
, then
(3.3)
where ![](https://www.scirp.org/html/15-7401087\14c6d05e-9e94-4a5b-8906-ea76900c18f8.jpg)
Proof. Using Lemma 2.1, the s-convexity of
on
, and Hölder’s integral inequality yields
![](https://www.scirp.org/html/15-7401087\e5a2dd69-28d2-45f8-b06a-726cdff208f0.jpg)
where an easy calculation gives
(3.4)
and
(3.5)
Substituting Equations (3.4) and (3.5) into the above inequality results in the inequality (3.3). The proof of Theorem 3.2 is complete.
Corollary 3.2.1. Under conditions of Theorem 3.2, if
, then
![](https://www.scirp.org/html/15-7401087\6ebc50f4-1c89-4d90-8163-09b6f83aa7db.jpg)
Theorem 3.3. Under conditions of Theorem 3.2, we have
(3.6)
Proof. Making use of Lemma 2.1, the s-convexity of
on
, and Hölder’s integral inequality leads to
![](https://www.scirp.org/html/15-7401087\f2e4a94f-c5a3-4989-a413-59ae15d8514b.jpg)
where
(3.7)
and
(3.8)
Substituting Equations (3.7) and (3.8) into the above inequality derives the inequality (3.6). The proof of Theorem 3.3 is complete.
Corollary 3.3.1. Under conditions of Theorem 3.3, if s = 1, then
![](https://www.scirp.org/html/15-7401087\d2ae50dd-cd3f-4931-9efc-bca44240a233.jpg)
Theorem 3.4. Under conditions of Theorem 3.2, we have
![](https://www.scirp.org/html/15-7401087\7c77bb65-a8eb-4fb9-90e0-c9b51b09136f.jpg)
Proof. Since
is s-convex on
, by Lemma 2.1 and Hölder’s inequality, we have
![](https://www.scirp.org/html/15-7401087\fb380424-96aa-4eb1-b332-fd06076b33c1.jpg)
and
![](https://www.scirp.org/html/15-7401087\3e1922be-2a64-47ea-bcfa-3aed2c066390.jpg)
where a straightforward computation gives
![](https://www.scirp.org/html/15-7401087\71767900-d334-4524-92d0-f22d47b979d1.jpg)
![](https://www.scirp.org/html/15-7401087\3c688def-9e56-4f69-8088-a718e6f275ce.jpg)
![](https://www.scirp.org/html/15-7401087\a4f8f851-981c-4d7b-b96f-ceb4475834a4.jpg)
![](https://www.scirp.org/html/15-7401087\792c8885-6372-4bd2-8270-d46ea6854373.jpg)
Substituting these equalities into the above inequality brings out the inequality (3.10). The proof of Theorem 3.4 is complete.
Corollary 3.4.1. Under conditions of Theorem 3.4, if
, then
![](https://www.scirp.org/html/15-7401087\7390b574-7a46-4945-8bbb-23b9716e2e6a.jpg)
4. Applications to Special Means
For positive numbers
and
, define
(4.1)
and
(4.2)
It is well known that A and
are respectively called the arithmetic and generalized logarithmic means of two positive number
and
.
Now we are in a position to construct some inequalities for special means A and
by applying the above established inequalities of Hermite-Hadamard type.
Let
(4.3)
for
and
. Since
and
![](https://www.scirp.org/html/15-7401087\855f1ff8-1fb8-4b5f-8fc9-b1d656bc3ff4.jpg)
for
and
then
is s-convex function on
and
![](https://www.scirp.org/html/15-7401087\df31e624-340f-4972-b8b4-209f683640b7.jpg)
![](https://www.scirp.org/html/15-7401087\dc9a34e4-b0df-4f70-8dea-ab55ba4fad7a.jpg)
![](https://www.scirp.org/html/15-7401087\196f8d32-2188-49d7-bdc4-42d46f03f9cf.jpg)
Applying the function (4.3) to Theorems 3.1 to 3.3 immediately leads to the following inequalities involving special means
and
.
Theorem 4.1. Let
, and
. Then
![](https://www.scirp.org/html/15-7401087\3f77a233-b6a9-4e3c-8e30-3d98ed53584a.jpg)
Theorem 4.2. For
,
, and
, we have
(4.4)
Theorem 4.3. For
,
, and
, we have
![](https://www.scirp.org/html/15-7401087\62a4f932-e331-4a74-a1bf-b663f6d28909.jpg)
5. Acknowledgements
The first author was supported by Science Research Funding of Inner Mongolia University for Nationalities under Grant No. NMD1103.