1. Introduction

The following inequality is well known in the literature as Simpson’s inequality.

Theorem 1. Let be a four times continuously differentiable mapping on and Then, the following inequality holds:

Interested readers may find useful details on Simpson’s type of integral inequalities in (^{[1, 4, 6]}).

We recall that the notion of quasi-convex functions generalized the notion of convex functions. More precisely, a function is said to be quasi-convex on if

Clearly, any convex function is a quasi-convex function but the reverse are not true. Because there exist quasi-convex functions which are not convex, (see for instance ^{[2, 3, 5, 8]} )

The following Simpson’s type inequalities for quasi-convex functions was obtained proved by Set et.al in ^[9].

Theorem 2. Let be a differentiable mapping on such that wherewith If is quasi-convex on then the following inequality holds:

(1.1)

Theorem 3. Let be a differentiable mapping on such that where with If is quasi-convex on and then the following inequality holds:

(1.2)

where

Theorem 4. Let be a differentiable mapping on such that where with If is quasi-convex on and then the following inequality holds:

(1.3)

The main aim of this paper is to establish the generalization of Simpson’s type inequalities obtained by Set et. al. for the class of functions whose derivatives in absolute value at certain powers are quasi-convex functions.

2. Main Results

We need the following integral identity which plays a key role in order to prove our main theorems, see ^[7].

Lemma 1. Let be an absolutely continuous mapping on wherewith If then for

where

A simple proof of this equality can be also done by integrating by parts in the right hand side. The details are left to the interested reader.

Theorem 5. Let be a differentiable mapping on and where with If is quasi-convex function, then

(2.1)

Proof. Using Lemma 1, and the fact that is quasi-convex, we have

where we use the fact that

which completes the proof.

Remark 1. If we choose in (2.1), then we get (1.1).

A similar result is embodied in the following theorem.

Theorem 6. Let be a differentiable mapping on and where with If is quasi-convex function, then

(2.2)

where and .

Proof. Using Lemma 1, well known Hölder’s integral inequality and the fact that is quasi-convex, we have

where we use the fact that

which completes the proof.

Remark 2. If we choose n = 1 in (2.2), then we get (1.2).

A more general inequality is given as follows.

Theorem 7. Let be a differentiable mapping on and where with If is quasi-convex function and then

(2.3)

Proof. Suppose that . From Lemma 1 and using the well known power mean inequality, we have

Therefore, the proof is completed.

Remark 3. Theorem 7 is equal to Theorem 5 for

Remark 4. If we choose in (2.3), then we get (1.3).

3. Applications to Special Means

In the literature, the following means for real numbers are well known:

(a) The arithmetic mean:

(b) The logarithmic mean:

(c) The p-logarithmic mean:

Now we shall use the results of Section 2 to prove the following new facts for the above means.

Proposition 1. Let and Then, we have

Proof. The assertion follows from Theorem 5 applied to the quasi-convex mapping and

Proposition 2. Let and Then, we have

Proof. The assertion follows from Theorem 6 applied to the quasi-convex mapping and

Proposition 3. Let and Then, for all we have

Proof. The assertion follows from Theorem 7 applied to the quasi-convex mapping and

References

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