Trapezoidal Type Fejér Inequalities Related to Harmonically Convex Functions and Application

Abstract

Some authors introduced the concepts of the harmonically arithmetic convex functions and establish some integral inequalities of Hermite Hadamard Fejér type related to the harmonically arithmetic convex functions. In this paper, a mapping is considered to get some preliminary results and a new trapezoidal form of Fejér inequality related to the harmonically arithmetic convex functions. By using a mapping , the new theorems and corollaries are obtained. Taking advantage of these, applications were given for some real number averages.

1. Introduction

In 1906, the Hungarian mathematician L. Fejér proved the following integral inequalities known in the literature as Fejér inequality [1, 2]:where is convex and is integrable and symmetric to . For some other inequalities in connection with Fejér inequalities see [3–8] and the references therein.

In [9], Hwang found out the Fejér trapezoidal inequality related to convex functions as follows:

Theorem 1. Let be a differentiable mapping, where with , and let be a continuous positive mapping symmetric to . If the mapping is convex on , then the following inequality holds:

In [8], Sarıkaya revealed new lemma and the difference between the right and middle part of (1) using Hölder’s inequality for convex function as follows:

Lemma 2 (see [8]). Let be a differentiable mapping on , with , and let be a differentiable mapping. If , then the following equality holds:for each , where

Theorem 3 (see [8]). Let be a differentiable mapping, with , and let be a differentiable mapping symmetric to . If is convex on , , then the following inequality holds: where and

Theorem 4 ([1] Theorem 2.1). Let be a mapping that is differentiable on , let be points with , and let be a nonnegative integrable mapping that is differentiable on . If is symmetric to and if is convex on , then

In [6], İşcan identified the harmonically convex function and proved Hermite Hadamard type inequality connected with harmonically convex function as follows:

Definition 5 (see [6]). Let be a real interval. A function is said to be harmonically convex iffor and . If the inequalities in (1) are reversed then is said to be harmonically concave.

Theorem 6 (see [6]). Let be a harmonically convex function and with . If then the following inequalities hold

Theorem 7 (see [6]). Let be differentiable function on , , with and . If is harmonically convex on for , thenwhere

Definition 8 (see [10]). A function is said to satisfy Lipschitz condition on if there is a constant so that for any two points ,

In this paper, I obtain a new trapezoidal form of Fejér inequality via the absolute value of the derivative of the considered function is the harmonically convex function. In addition, I get features of mapping as lemma and the new theorems and new corollaries. Furthermore, some applications in connection with special means are given.

2. Main Results

In the section, we have obtained the new theorem and corollary about Hermite Hadamard Fejér type inequality for the both harmonically convex functions.

We use this lemma for harmonically convex function and motivated by above works and results we consider a mapping and obtain some introductory properties related to it. Also a new trapezoidal form of Fejér inequality is proved in the case that the absolute value of considered function is harmonically convex.

Related to a function consider the mapping as the following:There exist some properties for the mapping , compiled in the following lemma which are used to obtain our main results.

Lemma 9. Suppose that is an interval, with and is an integrable function on .(1)If is symmetric to , then(2)For any , (3)If is a nonnegative function, then(4)The following inequalities hold. and(5)Let be an interval. is a differentiable mapping on , is an interior of , and is a differentiable nonnegative mapping. If , then we get the following equality:

Proof. (1) By taking the change of variable to (13), for , we obtainwhere . Because of being symmetric to ,and soHoweverIf we use (22) and (23) to (20), we getwhere .
By using the same argument as above, we can prove thatwhere .
(2) It is easy consequence of assertion of (1)
(3) By the assertion (3), we can get the following relations: For the second part of (4), we conceive the following assertion which is not hard to prove:By using Hölder inequality to the last inequality we getNow using (29) in (27) inequality, we get(4) Firstly we calculate the equality as follows: If we pulse both of the last equal with , then the proof is completed.

Theorem 10. Suppose that is a differentiable mapping on , with and is a differentiable mapping. Assume that is an integrable on and there exist constants such thatfor all . Then

Proof. From (5) of Lemma 9, we get as a result of If it is used the absolute value to both sides of the last equality, we have since from the inequality , we have which implies that

Remark 11. If we take is symmetric to , then from Lemma 9, we get and

Theorem 12. Suppose that is a differentiable mapping on , with and is a differentiable mapping symmetric to . If is a harmonically convex mapping on , thenwhereand

Proof. From the definition of , claim of Lemma 9, and being a harmonically convex functions, we haveIf we change the order of integration, we getSince function is symmetric to , thenAlso it is not hard to see thatandBy using (45), (46), and (47) to (44), then we complete this proof.

Theorem 13. Let be a mapping differentiable on , let be points w,th , and let be a nonnegative integrable mapping that is differentiable on . Assume that is integrable on and satisfies a Lipschitz condition for some . Then

Proof. By using (5) of Lemma 9, we have From the last equality, we get If satisfies a Lipschitz condition as (12) for some , then Because of this inequality, the proof is completed.

Remark 14. In Theorem 13, suppose that is symmetric to . If we use (1) of Lemma 9, we obtainwhich implies that

Corollary 15. In Theorem 13, if we take , for , then

3. Application

Recall the following means which could be considered extensions of arithmetic, geometric, harmonic, and generalized logarithmic from positive to real numbers.(1)The arithmetic mean:(2)The geometric mean: (3)The harmonic mean: (4)The generalized logarithmic mean: Now, we use these means in Corollary 15, Theorem 12 and Remark 11.

Proposition 16. Let , , and . Then

Proof. The claim follows from Corollary 15 for . Since is a convex and nondecreasing function, is a harmonically convex function.

Proposition 17. Let , and . Thenwhereand

Proof. The claim follows from Theorem 12 for and . Since is a convex and nondecreasing function, is a harmonically convex function.

Proposition 18. Let , and . Then

Proof. The claim follows from Remark 11 for and . Since is a convex and nondecreasing function, is a harmonically convex function.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that they have no conflicts of interest.