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Journal of Inequalities and Applications
 
J Inequal Appl. 2017; 2017(1): 169.
Published online 2017 July 19. doi:  10.1186/s13660-017-1444-6
PMCID: PMC5517584

Generalized Hermite-Hadamard type inequalities involving fractional integral operators

Abstract

In this article, a new general integral identity involving generalized fractional integral operators is established. With the help of this identity new Hermite-Hadamard type inequalities are obtained for functions whose absolute values of derivatives are convex. As a consequence, the main results of this paper generalize the existing Hermite-Hadamard type inequalities involving the Riemann-Liouville fractional integral.

Keywords: Hermite-Hadamard inequality, convex function, Hölder inequality, fractional integral operator

Introduction and preliminaries

During the last century the theory of convexity has emerged as an interesting and fascinating field of mathematics. It plays a pivotal role in optimization theory, functional analysis, control theory and economics etc.

A function f:I ⊆ ℝ → ℝ is said to be convex if the inequality

f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y)

holds for all xy ∈ I and t ∈ [0, 1].

The following inequality is a so-called classical Hermite-Hadamard type inequality for convex functions. Let f:I = [ab] ⊆ ℝ → ℝ be a convex function and ab ∈ I with a < b, then

f(a+b2)1baabf(x)dxf(a)+f(b)2.
1.1

This inequality is one of the most useful inequalities in mathematical analysis. For new proofs, noteworthy extension, generalizations and numerous applications on this inequality, see, e.g., [13] where further references are given.

The relationship between theory of convexity and theory of inequalities has motivated many researchers to study these theories in depth. As a consequence of this fact several inequalities have been obtained via convex functions; see [1].

The history of fractional calculus can be traced back to the letter of L’Hospital to Leibniz in which he inquired him about the notation he was using for the nth derivative of the linear function f(x) = x, DnxDxn. L’Hospital asked the question: what would the result be if n = ½. Leibniz replied: An apparent paradox, from which one day useful consequences will be drawn. Nowadays fractional calculus has become a powerful tool in many branches of mathematics. Sarikaya et al. [4] used the definitions of Riemannn-Liouville integrals and developed a new generalization of Hermite-Hadamard inequality. This result inspired many researchers to study this area. For more details, and for recent results and recently found properties concerning this operator one can consult [412].

We need some definition and mathematical preliminaries of fractional calculus theory for using in this study as follows.

Definition 1.1

Let f ∈ L[ab]. The Riemann-Liouville integrals Ja+αf and Jbαf of order α > 0 with a ≥ 0 are defined by

Ja+αf(x)=1Γ(α)ax(xt)α1f(t)dt,x>a,

and

Jbαf(x)=1Γ(α)xb(tx)α1f(t)dt,x<b,

respectively. Here Γ(t) is the Gamma function and its definition is Γ(t)=0exxt1dx. It is to be noted that Ja+0f(x)=Jb0f(x)=f(x); in the case of α = 1, the fractional integral reduces to the classical integral.

In [13], Zhu et al. established a new identity for differentiable convex mappings via the Riemann-Liouville fractional integral.

Lemma 1.1

[13]

Let f:[ab] → ℝ be a differentiable mapping on (ab) with a < b. If f ∈ L[ab], then the following equality for fractional integrals hold:

Γ(α+1)2(ba)α[(Jαaf)(b)+(Jα+bf)(a)]f(a+b2)=ba2[01k(t)f(ta+(1t)b)dt01[(1t)αtα]f(ta+(1t)b)dt],
1.2

where

k(t)={1,0<t12,1,12<t1.

Using the above identity, they gave the following result for the Riemann-Liouville fractional integral.

Theorem 1.1

[13]

Let f:[ab] → ℝ be a differentiable mapping on (ab) with a < b. If |f| is convex on [ab], then the following fractional inequality for fractional integrals holds:

|Γ(α+1)2(ba)α[(Jαaf)(b)+(Jα+bf)(a)]f(a+b2)|ba4(α+1)(α+312α1)[|f(a)|+|f(b)|].
1.3

In [14], Raina introduced a class of functions defined formally by

Fρ,λσ(x)=Fρ,λσ(0),σ(1),(x)=k=0σ(k)Γ(ρk+λ)xk(ρ,λ>0;|x|<R),
1.4

where the coefficients σ(k) (k ∈ ℕ = ℕ ∪ {0}) are a bounded sequence of positive real numbers and R is the set of real numbers. With the help of (1.4), Raina [14] and Agarwal et al. [15] defined the following left-sided and right-sided fractional integral operators, respectively:

(Jρ,λ,a+;wσφ)(x)=ax(xt)λ1Fρ,λσ[w(xt)ρ]φ(t)dt(x>a>0),
1.5

(Jρ,λ,b;wσφ)(x)=xb(tx)λ1Fρ,λσ[w(tx)ρ]φ(t)dt(0<x<b),
1.6

where λρ > 0, w ∈ ℝ and φ(t) is such that the integral on the right side exists. Recently some new integral inequalities involving this operator have appeared in the literature (see, e.g., [1522]).

It is easy to verify that Jρ,λ,a+;wσφ(x) and Jρ,λ,b;wσφ(x) are bounded integral operators on L(ab), if

M:=Fρ,λ+1σ[w(ba)ρ]<.
1.7

In fact, for φ ∈ L(ab), we have

Jρ,λ,a+;wσφ(x)1M(ba)λφ1
1.8

and

Jρ,λ,b;wσφ(x)1M(ba)λφ1,
1.9

where

φp:=(ab|φ(t)|pdt)1p.

Here, many useful fractional integral operators can be obtained by specializing the coefficient σ(k). For instance the classical Riemann-Liouville fractional integrals Ja+α and Jbα of order α follow easily by setting λα, σ(0) = 1 and w = 0 in (1.5) and (1.6).

Motivated by the work in [1315], firstly, we will prove a generalization of the identity given by Zhu et al. using generalized fractional integral operators. Then we will give some new Hermite-Hadamard type inequalities, which are generalizations of the results in [13] to the case λα, σ(0) = 1 and w = 0. Our results can be viewed as a significant extension and generalization of the previously known results.

Results and discussions

In this section, we derive our main results. For the sake of simplicity, we denote

Lf(a,b;w;J):=12(ba)λ[(Jρ,λ,b;wσf)(a)+(Jρ,λ,a+;wσf)(b)]Fρ,λ+1σ[w(ba)ρ]f(a+b2).

Lemma 2.1

Let f:[ab] → ℝ be a differentiable mapping on (ab) with a < b. If f ∈ L[ab], then the following equality for generalized fractional integral operators holds:

Lf(a,b;w;J)=(ba)2{01k(t)f(ta+(1t)b)dt01(1t)λFρ,λ+1σ[w(ba)ρ(1t)ρ]f(ta+(1t)b)dt+01tλFρ,λ+1σ[w(ba)ρtρ]f(ta+(1t)b)dt},

where

k(t)={Fρ,λ+1σ[w(ba)ρ],0<t12,Fρ,λ+1σ[w(ba)ρ],12<t1,

ρλ > 0, w ∈ ℝ.

Proof

It suffices to note that

I=012Fρ,λ+1σ[w(ba)ρ]f(ta+(1t)b)dt121Fρ,λ+1σ[w(ba)ρ]f(ta+(1t)b)dt01(1t)λFρ,λ+1σ[w(ba)ρ(1t)ρ]f(ta+(1t)b)dt+01tλFρ,λ+1σ[w(ba)ρtρ]f(ta+(1t)b)dt:=I1+I2+I3+I4.
2.1

Changing variables with xta + (1 − t)b, we get

I1=012Fρ,λ+1σ[w(ba)ρ]f(ta+(1t)b)dt=Fρ,λ+1σ[w(ba)ρ]ba[f(b)f(a+b2)],I2=121Fρ,λ+1σ[w(ba)ρ]f(ta+(1t)b)dt=Fρ,λ+1σ[w(ba)ρ]ba[f(a)f(a+b2)].

Integrating by parts, we have

I3=01(1t)λFρ,λ+1σ[w(ba)ρ(1t)ρ]f(ta+(1t)b)dt=1ba(1t)λFρ,λ+1σ[w(ba)ρ(1t)ρ]f(ta+(1t)b)|01+1ba01(1t)λ1Fρ,λσ[w(ba)ρ(1t)ρ]f(ta+(1t)b)dt=1baFρ,λ+1σ[w(ba)ρ]f(b)+1baab(xaba)λ1Fρ,λσ[w(ba)ρ(xaba)ρ]f(x)badx=1baFρ,λ+1σ[w(ba)ρ]f(b)+1(ba)λ+1(Jρ,λ,b;wσf)(a).
2.2

Analogously

I4=01tλFρ,λ+1σ[w(ba)ρtρ]f(ta+(1t)b)dt=1batλFρ,λ+1σ[w(ba)ρtρ]f(ta+(1t)b)|01+1ba01tλ1Fρ,λσ[w(ba)ρtρ]f(ta+(1t)b)dt=1baFρ,λ+1σ[w(ba)ρ]f(a)+1baab(bxba)λ1Fρ,λσ[w(ba)ρ(bxba)ρ]f(x)badx=1baFρ,λ+1σ[w(ba)ρ]f(a)+1(ba)λ+1(Jρ,λ,a+;wσf)(b).
2.3

Substituting the resulting equalities into equality (2.1), we have

I=2baFρ,λ+1σ[w(ba)ρ]f(a+b2)+1(ba)λ+1[(Jρ,λ,b;wσf)(a)+(Jρ,λ,a+;wσf)(b)].
2.4

Thus, multiplying both sides by (ba)2, the result is obtained.

Remark 2.1

Choosing λα, σ(0) = 1 and w = 0 in Lemma 2.1, equality (2.1) reduces to equality (1.2).

Theorem 2.1

Let f:[ab] → ℝ be a differentiable function on [ab] with a < b. If |f| is convex on (ab), then the following inequality for generalized fractional integral operators holds:

|Lf(a,b;w;J)|ba4Fρ,λ+2σ1[|w|(ba)ρ][|f(a)|+|f(b)|],
2.5

where

σ1(k)=σ(k)(λ+ρk+312λ+ρk1),

ρλ > 0, w ∈ ℝ, s ∈ (0, 1].

Proof

Using Lemma 2.1 and convexity of |f|, we have

|Lf(a,b;w;J)|ba2{|012Fρ,λ+1σ[w(ba)ρ]f(ta+(1t)b)dt+121Fρ,λ+1σ[w(ba)ρ]f(ta+(1t)b)dt|+|01(1t)λFρ,λ+1σ[w(ba)ρ(1t)ρ]f(ta+(1t)b)dt01tλFρ,λ+1σ[w(ba)ρtρ]f(ta+(1t)b)dt|}ba2{01|Fρ,λ+1σ[w(ba)ρ]|(t|f(a)|+(1t)|f(b)|)dt+k=0σ(k)|w|k(ba)ρkΓ(ρk+λ+1)[012((1t)λ+ρktλ+ρk)(t|f(a)|+(1t)|f(b)|)dt+121(tλ+ρk(1t)λ+ρk)(t|f(a)|+(1t)|f(b)|)dt]}ba2k=0σ(k)|w|k(ba)ρkΓ(ρk+λ+1)×{(|f(a)|+|f(b)|2)+|f(a)|012(t(1t)λ+ρktλ+ρk+1)dt+|f(b)|012((1t)λ+ρk+1tλ+ρk(1t))dt+|f(a)|121(tλ+ρk+1t(1t)λ+ρk)dt+|f(b)|121((1t)tλ+ρk(1t)λ+ρk+1)dt}=ba2k=0σ(k)|w|k(ba)ρkΓ(ρk+λ+1)[12+1λ+ρk+1(112λ+ρk)][|f(a)|+|f(b)|]=ba4k=0σ(k)|w|k(ba)ρkΓ(ρk+λ+2)(λ+ρk+312λ+ρk1)[|f(a)|+|f(b)|]=ba4Fρ,λ+2σ1[|w|(ba)ρ][|f(a)|+|f(b)|],
2.6

using the facts that

012(t(1t)λ+ρktλ+ρk+1)dt=121((1t)tλ+ρk(1t)λ+ρk+1)dt=2λ+ρk+1(λ+ρk+2)(λ+ρk+1)(λ+ρk+2)2λ+ρk+1

and

012((1t)λ+ρk+1tλ+ρk(1t))dt=121(tλ+ρk+1t(1t)λ+ρk)dt=1λ+ρk+21(λ+ρk+2)2λ+ρk+2λ+ρk+3(λ+ρk+1)(λ+ρk+2)2λ+ρk+2.

Thus the proof is completed.

Remark 2.2

Choosing λα, σ(0) = 1 and w = 0 in Theorem 2.1, inequality (2.5) reduces to inequality (1.3).

Theorem 2.2

Let f:[ab] → ℝ be a differentiable function on (ab) with a < b. If |f|q is convex and q > 1 with 1p+1q=1, then the following inequality for generalized fractional integral operators holds:

|Lf(a,b;w;J)|ba2[Fρ,λ+1σ[|w|(ba)ρ](|f(a)|q+|f(a)|q2)1q+Fρ,λ+1σ2[|w|(ba)ρ]([18|f(a)|q+38|f(b)|q]1q+[38|f(a)|q+18|f(b)|q]1q)]Fρ,λ+1σ3[|w|(ba)ρ][|f(a)|+|f(b)|],
2.7

where

σ2(k)=σ(k)[1p(λ+ρk)+1(112p(λ+ρk))]1p,σ3(k)=σ(k)(12)1q(1+[4p(λ+ρk)+1(112p(λ+ρk))]1p),

ρλ > 0 and w ∈ ℝ.

Proof

By using Lemma 2.1, we have

|Lf(a,b;w;J)|ba2{|012Fρ,λ+1σ[w(ba)ρ]f(ta+(1t)b)dt+121Fρ,λ+1σ[w(ba)ρ]f(ta+(1t)b)dt|+|01(1t)λFρ,λ+1σ[w(ba)ρ(1t)ρ]f(ta+(1t)b)dt01tλFρ,λ+1σ[w(ba)ρtρ]f(ta+(1t)b)dt|}ba2{k=0σ(k)|w|k(ba)ρkΓ(ρk+λ+1)01|f(ta+(1t)b)|dt+k=0σ(k)|w|k(ba)ρkΓ(ρk+λ+1)[012((1t)λ+ρktλ+ρk)|f(ta+(1t)b)|dt+121(tλ+ρk(1t)λ+ρk)|f(ta+(1t)b)|dt]}.
2.8

Using the well-known Hölder inequality and convexity of |f|q we get

01|f(ta+(1t)b)|dt(|f(a)|q+|f(a)|q2)1q.
2.9

Thus

012((1t)λ+ρktλ+ρk)|f(ta+(1t)b)|dt[012((1t)λ+ρktλ+ρk)pdt]1p[012|f(ta+(1t)b)|qdt]1q[012((1t)p(λ+ρk)tp(λ+ρk))dt]1p[012(t|f(a)|q+(1t)|f(a)|q)dt]1q=[1p(λ+ρk)+1(112p(λ+ρk))]1p[18|f(a)|q+38|f(b)|q]1q
2.10

and

121(tλ+ρk(1t)λ+ρk)|f(ta+(1t)b)|dt[121(tλ+ρk(1t)λ+ρk)pdt]1p[012|f(ta+(1t)b)|qdt]1q[121(tp(λ+ρk)(1t)p(λ+ρk))dt]1p[121(t|f(a)|q+(1t)|f(a)|q)dt]1q=[1p(λ+ρk)+1(112p(λ+ρk))]1p[38|f(a)|q+18|f(b)|q]1q,
2.11

where we used that (AB)p ≤ Ap − Bp for any A ≥ B ≥ 0 and p ≥ 1 in (2.10) and (2.11).

Let a1 = 3|f(a)|q, b1 = |f(b)|q, a2 = |f(a)|q, b2 = 3|f(b)|q. Here 0<1q<1 for q > 1. We use the fact that

k=1n(ak+bk)sk=1naks+k=1nbks.

For 0 ≤ s < 1, a1a2a3, …, an ≥ 0, b1b2b3, …, bn ≥ 0. Combining the inequalities (2.10) with (2.11) we obtain

012((1t)λ+ρktλ+ρk)|f(ta+(1t)b)|dt+121(tλ+ρk(1t)λ+ρk)|f(ta+(1t)b)|dt[1p(λ+ρk)+1(112p(λ+ρk))]1p(18)1q×([3|f(a)|q+|f(b)|q]1q+[|f(a)|q+3|f(b)|q]1q)[1p(λ+ρk)+1(112p(λ+ρk))]1p(18)1q(31q+1)[|f(a)|+|f(b)|][1p(λ+ρk)+1(112p(λ+ρk))]1p(18)1q4[|f(a)|+|f(b)|]=[4p(λ+ρk)+1(112p(λ+ρk))]1p(12)1q[|f(a)|+|f(b)|]
2.12

and

(|f(a)|q+|f(a)|q2)1q(12)1q[|f(a)|+|f(b)|].
2.13

Thus putting the inequalities (2.9), (2.12) and (2.13) in (2.8), the proof is completed.

Corollary 2.1

Choosing λα, σ(0) = 1 and w = 0 in Theorem  2.2, inequality (2.7) becomes the following inequality:

|Γ(α+1)2(ba)α[(Jαaf)(b)+(Jα+bf)(a)]f(a+b2)|ba2{[|f(a)|q+|f(a)|q2]1q+[1αp+1(112αp)]1p([18|f(a)|q+38|f(b)|q]1q+[38|f(a)|q+18|f(b)|q]1q)}ba2(1+[4αp+1(112αp)]1p)(12)1q[|f(a)|+|f(b)|].
2.14

Conclusion

In this paper, we have obtained a new fractional integral identity. Utilizing this new identity as an auxiliary result, we have obtained some new variants of Hermite-Hadamard type inequalities. The results derived in this paper become natural generalizations of classical results. It is expected that the interested reader may find useful applications of these results and consequently this paper may stimulate further research in this area.

Acknowledgements

The authors are thankful to the anonymous referee for his/her valuable comments and suggestions. The authors are pleased to acknowledge the support of Distinguished Scientist Fellowship Program (DSFP), King Saud University, Riyadh, Saudi Arabia. Third author is thankful to HEC, Pakistan for SRGP project 21-985/SRGP/R&D/HEC/2016.

Footnotes

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ES, MAN, MUA and AG worked jointly. All the authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Contributor Information

Erhan Set, moc.oohay@tesnahre.

Muhammed Aslam Noor, as.ude.usk@malsamroon.

Muhammed Uzair Awan, moc.liamg@riazu.nawa.

Abdurrahman Gözpinar, moc.liamg@97ranipzognamharrudba.

References

1. Dragomir SS, Pearce CEM. Selected Topics on Hermite-Hadamard Inequalities and Applications. 2000.
2. Mitrinović DS, Lacković IB. Hermite and convexity. Aequ. Math. 1985;28:229–232. doi: 10.1007/BF02189414. [Cross Ref]
3. Set E, Özdemir ME, Sarıkaya MZ. Inequalities of Hermite-Hadamard’s type for functions whose derivatives absolute values are m-convex. AIP Conf. Proc. 2010;1309(1):861–873. doi: 10.1063/1.3525219. [Cross Ref]
4. Sarıkaya MZ, Set E, Yaldız H, Başak N. Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 2013;57:2403–2407. doi: 10.1016/j.mcm.2011.12.048. [Cross Ref]
5. Dahmani Z. New inequalities in fractional integrals. Int. J. Nonlinear Sci. 2010;9(4):493–497.
6. Dahmani Z, Tabharit L, Taf S. New generalizations of Grüss inequality using Riemann-Liouville fractional integrals. Bull. Math. Anal. Appl. 2010;2(3):93–99.
7. Gorenflo R, Mainardi F. Fractals and Fractional Calculus in Continuum Mechanics. Wien: Springer; 1997. Fractional calculus: integral and differential equations of fractional order; pp. 223–276.
8. Noor MA, Cristescu G, Awan MU. Generalized fractional Hermite-Hadamard inequalities for twice differentiable s-convex functions. Filomat. 2015;29(4):807–815. doi: 10.2298/FIL1504807N. [Cross Ref]
9. Sarıkaya MZ, Yıldı rım H. On Hermite-Hadamard type inequalities for Riemannn-Liouville fractional integrals. Miskolc Math. Notes. 2016;17(2):1049–1059. doi: 10.18514/MMN.2017.1197. [Cross Ref]
10. Set E, Sarıkaya MZ, Özdemir ME, Yıldı rım H. The Hermite-Hadamard’s inequality for some convex functions via fractional integrals and related results. J. Appl. Math. Stat. Inform. 2014;10(2):69–83.
11. Set E. New inequalities of Ostrowski type for mapping whose derivatives are s-convex in the second sense via fractional integrals. Comput. Math. Appl. 2012;63:1147–1154. doi: 10.1016/j.camwa.2011.12.023. [Cross Ref]
12. Set E, İşcan İ, Sarıkaya MZ, Özdemir ME. On new inequalities of Hermite-Hadamard-Fejer type for convex functions via fractional integrals. Appl. Math. Comput. 2015;259:875–881. [PMC free article] [PubMed]
13. Zhu C, Feckan M, Wang J. Fractional integral inequalities for differentiable convex mappings and applications to special means and a midpoint formula. J. Appl. Math. Stat. Inform. 2012;8(2):21–28.
14. Raina RK. On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian Math. J. 2005;21(2):191–203.
15. Agarwal RP, Luo M-J, Raina RK. On Ostrowski type inequalities. Fasc. Math. 2016;204:5–27.
16. Set E, Gözpınar A. Some new inequalities involving generalized fractional integral operators for several class of functions. AIP Conf. Proc. 2017;1833 doi: 10.1063/1.4981686. [Cross Ref]
17. Set, E, Gözpınar, A: Hermite-Hadamard type inequalities for convex functions via generalized fractional integral operators. ResearchGate. https://www.researchgate.net/publication/312378686
18. Set, E, Akdemir, AO, Çelik, B: On generalization of Fejér type inequalities via fractional integral operator. ResearchGate. https://www.researchgate.net/publication/311452467
19. Set, E, Çelik, B: On generalization related to the left side of Fejér’s inequalites via fractional integral operator. ResearchGate. https://www.researchgate.net/publication/311651826
20. Set, E, Choi, J, Çelik, B: A new approach to generalized of Hermite-Hadamard inequality using fractional integral operator. ResearchGate. https://www.researchgate.net/publication/313437121
21. Usta, F, Budak, H, Sarıkaya, MZ, Set, E: On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators. Filomat (accepted)
22. Yaldız, H, Sarıkaya, MZ: On the Hermite-Hadamard type inequalities for fractional integral operator. ResearchGate. https://www.researchgate.net/publication/309824275

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