In the paper, the authors 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.
Integral Inequality; Hermite-Hadamard’s Integral Inequality; s-Convex Function; Derivative; Mean1. Introduction
The following definition is well known in the literature.
Definition 1.1. A function is said to be convex if
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
holds for all and.
Definition 1.3 ([2]). Let A function is said to be s-convex in the second sense if
for all and.
If is a convex function on with and, Then we have Hermite-Hardamard’s inequality
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
(2) If the new mapping is convex on
for, then
Theorem 1.2 ([4, Theorems 1 and 2]). Let be a differentiable function on and with, and let. If is convex on, then
and
Theorem 1.3 ([5, Theorems 2.3 and 2.4]). Let be differentiable on, with, and let. If is convex on, then
and
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
(2) If is s-convex on for some fixed and, then
Theorem 1.5 ([7, Theorem 2]). Let be an absolutely continuous function on such that for with. If is quasi-convex on, then
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
Proof. By integrating by part, we have
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
Proof. Since is s-convex on, by Lemma 2.1 and Hölder’s inequality, we have
where
and
Thus, we have
The proof of Theorem 3.1 is complete.
Corollary 3.1.1. Under conditions of Theorem 3.11) if, then
2) if, then
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
where
Proof. Using Lemma 2.1, the s-convexity of on, and Hölder’s integral inequality yields
where an easy calculation gives
and
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
Theorem 3.3. Under conditions of Theorem 3.2, we have
Proof. Making use of Lemma 2.1, the s-convexity of on, and Hölder’s integral inequality leads to
where
and
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
Theorem 3.4. Under conditions of Theorem 3.2, we have
Proof. Since is s-convex on, by Lemma 2.1 and Hölder’s inequality, we have
and
where a straightforward computation gives
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
4. Applications to Special Means
For positive numbers and, define
and
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
for and. Since and
for and then is s-convex function on and
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
Theorem 4.2. For, , and, we have
Theorem 4.3. For, , and, we have
5. Acknowledgements
The first author was supported by Science Research Funding of Inner Mongolia University for Nationalities under Grant No. NMD1103.
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