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

Mehmet Zeki Sarikaya*, Samet Erden

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

Keywords: Ostrowski’s inequality, Montgomery’s identities, convex function, Hölder inequality

Received June 17, 2014; Revised June 27, 2014; Accepted July 03, 2014

Citation: Mehmet Zeki Sarikaya, Samet Erden, On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function. Turkish Journal of Analysis and Number Theory, 2 (3). 85-89. January 2014. doi:10.12691/tjant-2-3-6

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]):

[picture] | (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:

[picture] | (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:

[picture] | (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:

[picture] | (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

[picture] | (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:

[picture] | (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:

[picture] | (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:

[picture] | (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:

[picture] | (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:

[picture] | (2.1)

where α > 1:

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

[picture] | (2.2)

and

[picture] | (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:

[picture] | (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:

[picture] | (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:

[picture] | (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]

[picture] | (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:

[picture] | (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].

References

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