On New Inequalities via Riemann-Liouville Fractional Integration

Abstract

We extend the Montgomery identities for the Riemann-Liouville fractional integrals. We also use these Montgomery identities to establish some new integral inequalities. Finally, we develop some integral inequalities for the fractional integral using differentiable convex functions.

1. Introduction

The inequality of Ostrowski [1] gives us an estimate for the deviation of the values of a smooth function from its mean value. More precisely, if 𝑓∶[𝑎,𝑏]→ℝ is a differentiable function with bounded derivative, then 𝑓(𝑥)−1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)≤1𝑑𝑡4+(𝑥−(𝑎+𝑏)/2)²(𝑏−𝑎)²(𝑏−𝑎)𝑓^′_∞, (1.1) for every 𝑥∈[𝑎,𝑏]. Moreover, the constant 1/4 is the best possible.

For some generalizations of this classic fact see ([2], (pages 468–484)) by Mitrinović et al. A simple proof of this fact can be done by using the following identity [2].

If 𝑓∶[𝑎,𝑏]→ℝ is differentiable on [𝑎,𝑏] with the first derivative 𝑓^′ integrable on [𝑎,𝑏], then Montgomery identity holds 𝑓(𝑥)=1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑑𝑡+^𝑏_𝑎𝑃₁(𝑥,𝑡)𝑓^′(𝑡)𝑑𝑡, (1.2) where 𝑃₁(𝑥,𝑡) is the Peano kernel defined by 𝑃₁(𝑥,𝑡)∶=𝑡−𝑎𝑏−𝑎,𝑎≤𝑡<𝑥,𝑡−𝑏𝑏−𝑎,𝑥≤𝑡≤𝑏. (1.3) Recently, several generalizations of the Ostrowski integral inequality are considered by many authors; for instance, covering the following concepts: functions of bounded variation, Lipschitzian, monotonic, absolutely continuous, and 𝑛-times differentiable mappings with error estimates with some special means together with some numerical quadrature rules. For recent results and generalizations concerning Ostrowski’s inequality, we refer the reader to the recent papers [3–10].

In this paper, we extend the Montgomery identities for the Riemann-Liouville fractional integrals. We also use these Montgomery identities to establish some new integral inequalities of Ostrowski's type. Finally, we develop some integral inequalities for the fractional integral using differentiable convex functions. Later, we develop some integral inequalities for the fractional integral using differentiable convex functions. From our results, the weighted and the classical Ostrowski's inequalities can be deduced as some special cases.

2. Fractional Calculus

Firstly, we give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used further in this paper. For more details, one can consult [11, 12].

Definition 2.1. The Riemann-Liouville fractional integral operator of order 𝛼≥0 with 𝑎≥0 is defined as 𝐽^𝛼_𝑎𝑓(𝑥)=1Γ(𝛼)^𝑥_𝑎(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝐽𝑑𝑡,_0𝑎𝑓(𝑥)=𝑓(𝑥). (2.1)

Recently, many authors have studied a number of inequalities by using the Riemann-Liouville fractional integrals, see [13–16] and the references cited therein.

3. Main Results

In order to prove some of our results, by using a different method of proof, we give the following identities, which are proved in [13]. Later, we will generalize the Montgomery identities in the next theorem.

Lemma 3.1. Let 𝑓∶𝐼⊂ℝ→ℝ be a differentiable function on 𝐼^∘ with 𝑎,𝑏∈𝐼(𝑎<𝑏) and 𝑓^′∈𝐿₁[𝑎,𝑏], then 𝑓(𝑥)=Γ(𝛼)()𝑏−𝑎𝑏−𝑥^1−𝛼𝐽^𝛼_𝑎𝑓(𝑏)−𝐽_𝑎^𝛼−1𝑃₂(𝑥,𝑏)𝑓(𝑏)+𝐽^𝛼_𝑎𝑃₂(𝑥,𝑏)𝑓^′(𝑏),𝛼≥1, (3.1) where 𝑃₂(𝑥,𝑡) is the fractional Peano kernel defined by 𝑃₂(𝑥,𝑡)∶=𝑡−𝑎()𝑏−𝑎𝑏−𝑥^1−𝛼Γ(𝛼),𝑎≤𝑡<𝑥,𝑡−𝑏()𝑏−𝑎𝑏−𝑥^1−𝛼Γ(𝛼),𝑥≤𝑡≤𝑏. (3.2)

Proof. By definition of 𝑃₁(𝑥,𝑡), we have Γ(𝛼)𝐽^𝛼_𝑎𝑃₁(𝑥,𝑏)𝑓^′(𝑏)=^𝑏_𝑎(𝑏−𝑡)^𝛼−1𝑃₁(𝑥,𝑡)𝑓^′(𝑡)=𝑑𝑡_𝑥𝑎(𝑏−𝑡)^𝛼−1𝑡−𝑎𝑓𝑏−𝑎^′(𝑡)𝑑𝑡+^𝑏_𝑥(𝑏−𝑡)^𝛼−1𝑡−𝑏𝑓𝑏−𝑎^′(𝑡)=1𝑑𝑡𝑏−𝑎^𝑥_𝑎(𝑏−𝑡)^𝛼−1(𝑡−𝑎)𝑓^′(𝑡)𝑑𝑡−^𝑏_𝑥(𝑏−𝑡)^𝛼𝑓^′(𝑡)=1𝑑𝑡𝐼𝑏−𝑎¹+𝐼₂. (3.3) Integrating by parts, we can state 𝐼₁=(𝑏−𝑡)^𝛼−1(𝑡−𝑎)𝑓(𝑡)^𝑥_𝑎−^𝑥_𝑎−(𝛼−1)(𝑏−𝑡)^𝛼−2(𝑡−𝑎)+(𝑏−𝑡)^𝛼−1𝑓(𝑡)=()𝑑𝑡𝑏−𝑥^𝛼−1(𝑥−𝑎)𝑓(𝑥)+(𝛼−1)^𝑥_𝑎(𝑏−𝑡)^𝛼−2(𝑡−𝑎)𝑓(𝑡)𝑑𝑡−^𝑥_𝑎(𝑏−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡, (3.4) and similarly, 𝐼₂=−(𝑏−𝑡)^𝛼𝑓(𝑡)^𝑏_𝑥−𝛼^𝑏_𝑥(𝑏−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡=(𝑏−𝑥)^𝛼𝑓(𝑥)−𝛼^𝑏_𝑥(𝑏−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡. (3.5) Adding (3.4) and (3.5), we get Γ(𝛼)𝐽^𝛼_𝑎𝑃₁(𝑥,𝑏)𝑓^′(𝑏)=1)𝑏−𝑎(𝑏−𝑎)(𝑏−𝑥^𝛼−1𝑓(𝑥)+(𝛼−1)^𝑥_𝑎(𝑏−𝑡)^𝛼−2(𝑡−𝑎)𝑓(𝑡)𝑑𝑡−𝛼^𝑏_𝑥(𝑏−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡−^𝑥_𝑎(𝑏−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡. (3.6) If we add and subtract the integral (𝛼−1)∫^𝑏_𝑥(𝑏−𝑡)^𝛼−2(𝑡−𝑏)𝑓(𝑡)𝑑𝑡 to the right-hand side of the equation above, then we have Γ(𝛼)𝐽^𝛼_𝑎𝑃₁(𝑥,𝑏)𝑓^′(𝑏)=1)𝑏−𝑎(𝑏−𝑎)(𝑏−𝑥^𝛼−1𝑓(𝑥)+(𝑏−𝑎)(𝛼−1)^𝑏_𝑎(𝑏−𝑡)^𝛼−2𝑃₁()𝑓(𝑡)−𝑥,𝑡𝑑𝑡_𝑏𝑎(𝑏−𝑡)^𝛼−1𝑓(𝑡)=()𝑑𝑡𝑏−𝑥^𝛼−1𝑓(𝑥)+(𝛼−1)^𝑏_𝑎(𝑏−𝑡)^𝛼−2𝑃₁()𝑓(𝑡)−1𝑥,𝑡𝑑𝑡𝑏−𝑎^𝑏_𝑎(𝑏−𝑡)^𝛼−1𝑓(𝑡)=()𝑑𝑡𝑏−𝑥^𝛼−1𝑓(𝑥)−Γ(𝛼)𝐽𝑏−𝑎^𝛼𝑎𝑓(𝑏)+Γ(𝛼)𝐽_𝑎^𝛼−1𝑃₁(𝑥,𝑏)𝑓(𝑏). (3.7) Multiplying both sides by (𝑏−𝑥)^1−𝛼, we obtain 𝐽^𝛼_𝑎𝑃₂(𝑥,𝑏)𝑓^′(𝑏)(𝑥)−Γ(𝛼)=𝑓()𝑏−𝑎𝑏−𝑥^1−𝛼𝐽^𝛼_𝑎𝑓(𝑏)+𝐽_𝑎^𝛼−1𝑃₂(𝑥,𝑏)𝑓(𝑏), (3.8) and so 𝑓(𝑥)=Γ(𝛼)()𝑏−𝑎𝑏−𝑥^1−𝛼𝐽^𝛼_𝑎𝑓(𝑏)−𝐽_𝑎^𝛼−1𝑃₂(𝑥,𝑏)𝑓(𝑏)+𝐽^𝛼_𝑎𝑃₂(𝑥,𝑏)𝑓^′(𝑏). (3.9) This completes the proof.

Now, we extend Lemma 3.1 as follows.

Theorem 3.2. Let 𝑓∶𝐼⊂ℝ→ℝ be a differentiable function on 𝐼^∘ with 𝑓^′∈𝐿₁[𝑎,𝑏], then the following identity holds: ()𝑓(𝑥)=Γ(𝛼)1−2𝜆()𝑏−𝑎𝑏−𝑥^1−𝛼𝐽^𝛼_𝑎𝑓(𝑏)−𝜆𝑏−𝑎𝑏−𝑥^𝛼−1𝑓(𝑎)−𝐽_𝑎^𝛼−1𝑃₃(𝑥,𝑏)𝑓(𝑏)+𝐽^𝛼_𝑎𝑃₃(𝑥,𝑏)𝑓^′(𝑏),𝛼≥1, (3.10) where 𝑃₃(𝑥,𝑡) is the fractional Peano kernel defined by 𝑃₃()()𝑥,𝑡∶=𝑡−1−𝜆𝑎−𝜆𝑏()𝑏−𝑎𝑏−𝑥^1−𝛼Γ(𝛼)(),𝑎≤𝑡<𝑥,𝑡−1−𝜆𝑏−𝜆𝑎()𝑏−𝑎𝑏−𝑥^1−𝛼Γ(𝛼),𝑥≤𝑡≤𝑏, (3.11) for 0≤𝜆≤1.

Proof. By similar way in proof of Lemma 3.1, we have Γ(𝛼)𝐽^𝛼_𝑎𝑃₃(𝑥,𝑏)𝑓^′(𝑏)=^𝑏_𝑎(𝑏−𝑡)^𝛼−1𝑃₃(𝑥,𝑡)𝑓^′(𝑡)=Γ(𝛼)𝑑𝑡)(𝑏−𝑥^1−𝛼𝑏−𝑎^𝑥_𝑎(𝑏−𝑡)^𝛼−1(𝑡−(1−𝜆)𝑎−𝜆𝑏)𝑓^′(𝑡)𝑑𝑡+^𝑏_𝑥(𝑏−𝑡)^𝛼−1(𝑡−(1−𝜆)𝑏−𝜆𝑎)𝑓^′(𝑡)=Γ(𝛼)𝑑𝑡)(𝑏−𝑥^1−𝛼𝐽𝑏−𝑎¹+𝐽₂. (3.12) Integrating by parts, we can state 𝐽₁=(𝑏−𝑥)^𝛼−1(𝑥−(1−𝜆)𝑎−𝜆𝑏)𝑓(𝑥)+(𝑏−𝑎)^𝛼𝑓(𝑎)+(𝛼−1)^𝑥_𝑎(𝑏−𝑡)^𝛼−2(𝑡−(1−𝜆)𝑎−𝜆𝑏)𝑓(𝑡)𝑑𝑡−^𝑥_𝑎(𝑏−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡, (3.13) and similarly, 𝐽₂=−(𝑏−𝑥)^𝛼(())𝑓(𝑥)+()𝑥−1−𝜆𝑏−𝜆𝑎𝛼−1^𝑏_𝑥(𝑏−𝑡)^𝛼−2(𝑡−(1−𝜆)𝑎−𝜆𝑏)𝑓(𝑡)𝑑𝑡−^𝑏_𝑥(𝑏−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡. (3.14) Thus, by using 𝐽₁ and 𝐽₂ in (3.12), we get (3.10) which completes the proof.

Remark 3.3. We note that in the special cases, if we take 𝜆=0 in Theorem 3.2, then we get (3.1) with the kernel 𝑃₂(𝑥,𝑡).

Theorem 3.4. Let 𝑓∶[𝑎,𝑏]→ℝ be a differentiable on (𝑎,𝑏) such that 𝑓^′∈𝐿₁[𝑎,𝑏], where 𝑎<𝑏. If |𝑓^′(𝑥)|≤𝑀 for every 𝑥∈[𝑎,𝑏] and 𝛼≥1, then the following inequality holds: )𝑓(𝑥)−Γ(𝛼)(1−2𝜆()𝑏−𝑎𝑏−𝑥^1−𝛼𝐽^𝛼_𝑎𝑓(𝑏)+𝜆𝑏−𝑎𝑏−𝑥^𝛼−1𝑓(𝑎)+𝐽_𝑎^𝛼−1𝑃₃()𝑓(𝑏)≤𝑀𝑥,𝑏𝛼())𝛼+1(𝑏−𝑎^𝛼(𝑏−𝑥)^1−𝛼2𝜆^𝛼+1+2(1−𝜆)^𝛼+1()+(+𝜆𝑏−𝑎−1𝑏−𝑥)2𝛼𝑏−𝑥−(.𝑏−𝑎𝛼+1) (3.15)

Proof. From Theorem 3.2, we get )𝑓(𝑥)−Γ(𝛼)(1−2𝜆()𝑏−𝑎𝑏−𝑥^1−𝛼𝐽^𝛼_𝑎𝑓(𝑏)+𝜆𝑏−𝑎𝑏−𝑥^𝛼−1𝑓(𝑎)+𝐽_𝑎^𝛼−1𝑃₃()𝑓(𝑏)≤1𝑥,𝑏Γ(𝛼)^𝑏_𝑎(𝑏−𝑡)^𝛼−1𝑃₃(𝑥,𝑡)𝑓^′(𝑡=())𝑑𝑡𝑏−𝑥^1−𝛼𝑏−𝑎^𝑥_𝑎(𝑏−𝑡)^𝛼−1|𝑡−(1−𝜆)𝑎−𝜆𝑏|𝑓^′(𝑡+)𝑑𝑡_𝑏𝑥(𝑏−𝑡)^𝛼−1|𝑡−(1−𝜆)𝑏−𝜆𝑎|𝑓^′(𝑡≤𝑀())𝑑𝑡𝑏−𝑥^1−𝛼𝑏−𝑎^𝑥_𝑎(𝑏−𝑡)^𝛼−1|𝑡−(1−𝜆)𝑎−𝜆𝑏|𝑑𝑡+^𝑏_𝑥(𝑏−𝑡)^𝛼−1|()|=𝑀()𝑡−1−𝜆𝑏−𝜆𝑎𝑑𝑡𝑏−𝑥^1−𝛼𝐽𝑏−𝑎³+𝐽₄. (3.16) By simple computation, we obtain 𝐽₃=^𝑥_𝑎(𝑏−𝑡)^𝛼−1|()|=𝑡−1−𝜆𝑎−𝜆𝑏𝑑𝑡_{𝑎𝜆𝑏+(1−𝜆)𝑎}(𝑏−𝑡)^𝛼−1(𝜆𝑏+(1−𝜆)𝑎−𝑡)𝑑𝑡+^𝑥_{𝜆𝑏+(1−𝜆)𝑎}(𝑏−𝑡)^𝛼−1(()𝑎)=()𝑡−𝜆𝑏−1−𝜆𝑑𝑡𝑏−𝑎^𝛼+1𝛼()2()𝛼+11−𝜆^𝛼+1()+()+𝜆𝑏−𝑎−1𝑏−𝑥^𝛼𝛼()[𝛼()−(,𝛼+1𝑏−𝑥1−𝜆)(𝑏−𝑎)(𝛼+1)] (3.17) and similarly 𝐽₄=^𝑏_𝑥(𝑏−𝑡)^𝛼−1|()|=𝑡−1−𝜆𝑏−𝜆𝑎𝑑𝑡_{𝑥𝜆𝑎+(1−𝜆)𝑏}(𝑏−𝑡)^𝛼−1(𝜆𝑎+(1−𝜆)𝑏−𝑡)𝑑𝑡+^𝑏_{𝜆𝑎+(1−𝜆)𝑏}(𝑏−𝑡)^𝛼−1(()𝑏)=𝑡−𝜆𝑎−1−𝜆𝑑𝑡2𝜆^𝛼+1(𝑏−𝑎)^𝛼+1𝛼()+()𝛼+1𝑏−𝑥^𝛼𝛼()[𝛼()(.𝛼+1𝑏−𝑥−𝜆𝑏−𝑎)(𝛼+1)] (3.18) By using 𝐽₃ and 𝐽₄ in (3.16), we obtain (3.15).

Remark 3.5. If we take 𝜆=0 in Theorem 3.4, then it reduces Theorem 4.1 proved by Anastassiou et al. [13]. So, our results are generalizations of the corresponding results of Anastassiou et al. [13].

Theorem 3.6. Let 𝑓∶[𝑎,𝑏]→ℝ be a differentiable convex function on (𝑎,𝑏) and 𝑓^′∈𝐿₁[𝑎,𝑏]. Then for any 𝑥∈(𝑎,𝑏), the following inequality holds: 1𝛼()𝛼()𝛼+1𝑏−𝑥²𝑓𝑏−𝑎^′+(𝑥)−(𝑏−𝑎)^𝛼(𝑏−𝑥)^1−𝛼()+𝛼𝑏−𝑥²−()𝑓𝑏−𝑎𝛼+1)(𝑏−𝑥^′₋(𝑥≤Γ(𝛼))()𝑏−𝑎𝑏−𝑥^1−𝛼𝐽^𝛼_𝑎𝑓(𝑏)−𝐽_𝑎^𝛼−1𝑃₂(𝑥,𝑏)𝑓(𝑏)−𝑓(𝑥),𝛼≥1. (3.19)

Proof. Similarly to the proof of Lemma 3.1, we have 𝑓(𝑥)−Γ(𝛼)()𝑏−𝑎𝑏−𝑥^1−𝛼𝐽^𝛼_𝑎𝑓(𝑏)+𝐽_𝑎^𝛼−1𝑃₂()𝑓(𝑏)=()𝑥,𝑏𝑏−𝑥^1−𝛼𝑏−𝑎^𝑥_𝑎(𝑏−𝑡)^𝛼−1(𝑡−𝑎)𝑓^′(𝑡)𝑑𝑡−^𝑏_𝑥(𝑏−𝑡)^𝛼𝑓^′(𝑡)𝑑𝑡. (3.20) Since 𝑓 is convex, then for any 𝑥∈(𝑎,𝑏) we have the following inequalities: 𝑓^′(𝑡)≤𝑓^′₋(𝑥)[],𝑓fora.e.𝑡∈𝑎,𝑥_′(𝑡)≥𝑓^′₊(𝑥)fora.e.𝑡∈[𝑥,𝑏]. (3.21)(3.22) If we multiply (3.21) by (𝑏−𝑡)^𝛼−1(𝑡−𝑎)≥0, 𝑡∈[𝑎,𝑥],𝛼≥1 and integrate on [𝑎,𝑥], we get ^𝑥_𝑎(𝑏−𝑡)^𝛼−1(𝑡−𝑎)𝑓^′(𝑡)𝑑𝑡≤^𝑥_𝑎(𝑏−𝑡)^𝛼−1(𝑡−𝑎)𝑓^′₋(𝑥)=1𝑑𝑡𝛼())𝛼+1(𝑏−𝑎^𝛼+1+(𝑏−𝑥)^𝛼[𝛼(𝑏−𝑥)−(𝛼+1)(𝑏−𝑎)]𝑓^′₋(𝑥), (3.23) and if we multiply (3.22) by (𝑏−𝑡)^𝛼≥0,𝑡∈[𝑥,𝑏],𝛼≥1 and integrate on [𝑥,𝑏], we also get ^𝑏_𝑥(𝑏−𝑡)^𝛼𝑓^′(𝑡)𝑑𝑡≥^𝑏_𝑥(𝑏−𝑡)^𝛼𝑓^′₊(𝑥)()𝑑𝑡=𝑏−𝑥^𝛼+1𝑓𝛼+1^′+(𝑥). (3.24) Finally, if we subtract (3.24) from (3.23) and use the representation (3.20) we deduce the desired inequality (3.19).

Corollary 3.7. Under the assumptions Theorem 3.6 with 𝛼=1, one has 12(𝑏−𝑥)²𝑓^′₊(𝑥)−(𝑎−𝑥)²𝑓^′₋(𝑥)≤^𝑏_𝑎𝑓(𝑡)𝑑𝑡−(𝑏−𝑎)𝑓(𝑥). (3.25)

The proof of Corollary 3.7 is proved by Dragomir in [6]. Hence, our results in Theorem 3.6 are generalizations of the corresponding results of Dragomir [6].

Remark 3.8. If we take 𝑥=(𝑎+𝑏)/2 in Corollary 3.7, we get 0≤𝑏−𝑎8𝑓^′₊𝑎+𝑏2−𝑓^′₋𝑎+𝑏2≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑑𝑡−𝑓𝑎+𝑏2. (3.26)

Theorem 3.9. Let 𝑓∶[𝑎,𝑏]→ℝ be a differentiable convex function on (𝑎,𝑏) and 𝑓^′∈𝐿₁[𝑎,𝑏]. Then for any 𝑥∈[𝑎,𝑏], the following inequality holds: Γ(𝛼)()𝑏−𝑎𝑏−𝑥^1−𝛼𝐽^𝛼_𝑎𝑓(𝑏)−𝐽_𝑎^𝛼−1𝑃₂()𝑓(𝑏)(𝑥)≤1𝑥,𝑏−𝑓𝛼()𝛼()𝛼+1𝑏−𝑥²𝑓𝑏−𝑎^′−(𝑏)−(𝑏−𝑎)^𝛼(𝑏−𝑥)^1−𝛼()+𝛼𝑏−𝑥²−(𝑏−𝑎𝛼+1)(𝑏−𝑥),𝛼≥1. (3.27)

Proof. Assume that 𝑓^′₊(𝑎) and 𝑓^′₋(𝑏) are finite. Since 𝑓 is convex on [𝑎,𝑏], then we have the following inequalities: 𝑓^′(𝑡)≥𝑓^′₊(𝑎)[],𝑓fora.e.𝑡∈𝑎,𝑥_′(𝑡)≤𝑓^′₋(𝑏)fora.e.𝑡∈[𝑥,𝑏]. (3.28)(3.29) If we multiply (3.28) by (𝑏−𝑡)^𝛼−1(𝑡−𝑎)≥0,𝑡∈[𝑎,𝑥],𝛼≥1 and integrate on [𝑎,𝑥], we have ^𝑥_𝑎(𝑏−𝑡)^𝛼−1(𝑡−𝑎)𝑓^′(𝑡)𝑑𝑡≥^𝑥_𝑎(𝑏−𝑡)^𝛼−1(𝑡−𝑎)𝑓^′₊(𝑎)=1𝑑𝑡𝛼())𝛼+1(𝑏−𝑎^𝛼+1+(𝑏−𝑥)^𝛼[𝛼(𝑏−𝑥)−(𝛼+1)(𝑏−𝑎)]𝑓^′₊(𝑎), (3.30) and if we multiply (3.29) by (𝑏−𝑡)^𝛼≥0,𝑡∈[𝑥,𝑏], 𝛼≥1 and integrate on [𝑥,𝑏], we also have ^𝑏_𝑥(𝑏−𝑡)^𝛼𝑓^′(𝑡)𝑑𝑡≤^𝑏_𝑥(𝑏−𝑡)^𝛼𝑓^′₋(𝑏)()𝑑𝑡=𝑏−𝑥^𝛼+1𝑓𝛼+1^′−(𝑏). (3.31) Finally, if we subtract (3.30) from (3.31) and use the representtation (3.20) we deduce the desired inequality (3.27).

Corollary 3.10. Under the assumptions Theorem 3.9 with 𝛼=1, one ^𝑏_𝑎𝑓(𝑡)()𝑓(𝑥)≤1𝑑𝑡−𝑏−𝑎2(𝑏−𝑥)²𝑓^′₋(𝑏)−(𝑎−𝑥)²𝑓^′₊(𝑎). (3.32)

The proof of Corollary 3.10 is proved by Dragomir in [6]. So, our results in Theorem 3.9 are generalizations of the corresponding results of Dragomir [6].

Remark 3.11. If we take 𝑥=(𝑎+𝑏)/2 in Corollary 3.10, we get 10≤𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑑𝑡−𝑓𝑎+𝑏2≤𝑏−𝑎8𝑓^′₋(𝑏)−𝑓^′₊(𝑎). (3.33)