﻿ On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function

### On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function

Mehmet Zeki Sarikaya, Samet Erden

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## On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function

Mehmet Zeki Sarikaya1,, Samet Erden2

1Department of Mathematics, Faculty of Science and Arts, Düzce University, Konu-ralp Campus, Düzce-TURKEY

2Department of Mathematics, Faculty of Science, Bartn University, Konuralp Cam-pus, BARTIN-TURKEY

### Abstract

In this paper, we extend some estimates of the right hand side of a Hermite- Hadamard-Fejér type inequality for functions whose first derivatives absolute values are convex. The results presented here would provide extensions of those given in earlier works.

• Sarikaya, Mehmet Zeki, and Samet Erden. "On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function." Turkish Journal of Analysis and Number Theory 2.3 (2014): 85-89.
• Sarikaya, M. Z. , & Erden, S. (2014). On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function. Turkish Journal of Analysis and Number Theory, 2(3), 85-89.
• Sarikaya, Mehmet Zeki, and Samet Erden. "On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function." Turkish Journal of Analysis and Number Theory 2, no. 3 (2014): 85-89.

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### 1. Introduction

Definition 1. The function , is said to be convex if the following inequality holds

for all x,y∈[a,b] and λ∈ [0,1]. We say that f is concave if (-f) is convex.

The following inequality is well known in the literature as the Hermite-Hadamard integral inequality (see, [4, 10]):

 (1.1)

where is a convex function on the interval I of real numbers and a, b ∈ I with a < b.

In [3], Dragomir and Agarwal proved the following results connected with the right part of (1.1).

Lemma 1. Let be a differentiable mapping on , with a < b. If f’ ∈ L[a,b], then the following equality holds:

 (1.2)

Theorem 1. Let be a differentiable mapping on, with a < b. If |f’| is convex on [a,b], then the following inequality holds:

 (1.3)

Theorem 2. Let be a differentiable mapping on , with a < b; f’ ∈ L(a,b) and p > 1. If the mapping |f’|p/(p-1) is convex on [a,b], then the following inequality holds:

 (1.4)

The most well-known inequalities related to the integral mean of a convex function are the Hermite Hadamard inequalities or its weighted versions, the so-called Hermite- Hadamard- Fejér inequalities (see, [8, 13, 14, 15, 16, 19, 20]). In [7], Fejer gave a weighted generalizatinon of the inequalities (1.1) as the following:

Theorem 3. , be a convex function, then the inequality

 (1.5)

holds, where is nonnegative, integrable, and symmetric about

In [13], some inequalities of Hermite-Hadamard-Fejer type for differentiable convex mappings were proved using the following lemma.

Lemma 2. Let be a differentiable mapping on , with a < b, and w: [a,b]→[0,∞) be a differentiable mapping. If f’ ∈L[a,b], then the following equality holds:

 (1.6)

for each t 2 [0,1]; where

The main result in [13] is as follows:

Theorem 4. Let be a differentiable mapping on , with a < b, and w: [a,b]→[0,∞) be a differentiable mapping and symmetric to. If |f’| is convex on [a,b] ; then the following inequality holds:

 (1.7)

where for t∈ [0,1].

Definition 2. Let f∈L1[a,b]. The Riemann-Liouville integrals of order α > 0 with a≥ 0 are defined by

And

respectively. Here, is the Gamma function and .

Meanwhile, Sarikaya et al. [12] presented the following important integral identity including the first-order derivative of f to establish many interesting Hermite-Hadamard type inequalities for convexity functions via Riemann-Liouville fractional integrals of the order α > 0.

Lemma 3. Let be a differentiable mapping on (a,b) with a < b. If f’∈L[a,b], then the following equality for fractional integrals holds:

 (1.8)

It is remarkable that Sarikaya et al. [12] first give the following interesting integral inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional

Theorem 5. Let be a positive function with 0 ≤ a < b and f∈L1 [a,b]. If f is a convex function on [a,b], then the following inequalities for fractional integrals hold:

 (1.9)

with α > 0:

For some recent results connected with fractional integral inequalities see [1, 2, 8, 17, 18].

In this article, using functions whose derivatives absolute values are convex, we obtained new inequalities of Hermite-Hadamard-Fejer type and Hermite-Hadamard type involving fractional integrals. The results presented here would provide extensions of those given in earlier works.

### 2. Main Results

We will establish some new results connected with the right-hand side of (1.5) and (1.1) involving fractional integrals used the following Lemma. Now, we give the following new Lemma for our results:

Lemma 4. Let be a differentiable mapping on , with a < b and let . If f’, w∈L[a,b], then, for all x∈ [a,b], the following equality holds:

 (2.1)

where α > 1:

Proof. By integration by parts, we have the following equalities:

 (2.2)

and

 (2.3)

Subtracting (2.3) from (2.2), we obtain (2.1). This completes the proof.

Remark 1. If we take w(s) = 1 in 2.1; the identity (2.1) reduces to the identity (1.8).

Corollary 1. Under the same assumptions of Lemma 4 with α = 1; then the following identity holds:

 (2.4)

Remark 2. If we take w(s) = 1 in (2.4), the identity (2.4) reduces to the identity (1.2).

Now, by using the above lemma, we prove our main theorems:

Theorem 6. Let be a differentiable mapping on , with a < b and let be continuous on [a,b]. If |f’| is convex on [a,b], then the following inequality holds:

where α > 0 and

Proof. We take absolute value of (2.1), we find that

Since |f’| is convex on [a,b], it follows that

Hence, the proof of theorem is completed.

Corollary 2. Under the same assumptions of Theorem 6 with w(s) = 1, then the following inequality holds:

 (2.5)

Proof. This proof is given by Sarikaya et. al in [11].

Remark 3. If we take α = 1 in (2.5); the inequality (2.5) reduces to (1.3).

Corollary 3. Under the same assumptions of Theorem 6 with α = 1, then the following inequality holds:

Theorem 7. Let be a differentiable mapping on , with a < b and let be continuous on [a,b]. If |f’|q is convex on [a,b], q > 1, then the following inequality holds:

 (2.6)

where α > 0, , and

Proof. We take absolute value of (2.1). Using Holder’s inequality, we find that

Since is convex on [a,b]

 (2.7)

From (2.7), it follows that

which this completes the proof.

Corollary 4. Under the same assumptions of Theorem 6 with w(s) = 1, then the following inequality holds:

 (2.8)

Corollary 5. Let the conditions of Theorem 7 hold. If we take α = 1 in (2.6), then the following inequality holds:

Remark 4. If we take w(s) = 1 in (2.9), we have

which is proved by Dragomir and Agarwal in [3].

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