Trapezoidal Type Fejér Inequalities Related to Harmonically Convex Functions and Application

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 mappingM(t) 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 M(t), the new theorems and corollaries are obtained. Taking advantage of these, applications were given for some real number averages.

Theorem 6 (see [6]). Let : ⊂ R \ {0} → R be a harmonically convex function and , ∈ with < . If ∈ [ , ] then the following inequalities hold Theorem 7 (see [6]). Let : ⊂ (0, ∞) → R be differentiable function on , , ∈ , with < and ∈ [ , ]. If | | is harmonically convex on [ , ] for ≥ 1, then Definition 8 (see [10] 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.

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 : [ , ] → R consider the mapping : [0, 1] → R as the following: There exist some properties for the mapping ( ), compiled in the following lemma which are used to obtain our main results.
(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: Journal of Function Spaces By using Hölder inequality to the last inequality we get Now using (29) in (27) inequality, we get (4) Firstly we calculate the equality as follows: If we pulse both of the last equal with −1/2, then the proof is completed.
If it is used the absolute value to both sides of the last equality, we have since from the inequality ≤ ( 2 2 /( + (1 − ) ) 2 ) ( / ( + (1 − ) )) ≤ , we have Remark 11. If we take is symmetric to 2 /( + ), then from Lemma 9, we get and   Proof. From the definition of ( ), claim of Lemma 9, and | | being a harmonically convex functions, we have If we change the order of integration, we get Also it is not hard to see that

Theorem 13. Let
: ⊂ R \ {0} → R be a mapping differentiable on , let , ∈ be points w,th < , and let : [ , ] → R be a nonnegative integrable mapping that is differentiable on ( , ). Assume that is integrable on [ , ] and satisfies a Lipschitz condition for some > 0. 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 > 0, then Because of this inequality, the proof is completed.

Application
Recall the following means which could be considered extensions of arithmetic, geometric, harmonic, and generalized logarithmic from positive to real numbers.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that they have no conflicts of interest.