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1 Turkih J Ineq, ) 7), Page 6 37 Turkih Journal of I N E Q U A L I T I E S Available online at wwwtjinequalitycom PARAMETERIZED HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS M ADIL KHAN AND TU KHAN Abtract The paper preent Hermite-Hadamard type inequalitie, which involve Riemann-Liouville fractional integral and contain an arbitrary parameter from the interval of definition of twice differentiable convex and concave function Introduction Fractional calculu i the notion of integral and derivative of arbitrary order, which i the generalization of integer-order differentiation and n-fold integration The beginning of fractional calculu i conidered to be the correpondence between L Hopital and Leibniz in 695, where the idea for differentiation of non-integer order wa dicued 9 Thi correpondence gave birth to the idea of fractional calculu Further contribution in thi area were made by Euler, Laplace, Fourier, Abel, Liouville, Riemann, Grunwald, Letnikov, Hadamard, Weyl, Riez, Marchaud, Kober and Caputo, 9, Fractional calculu play an important role in variou field uch a Electricity, Biology, Economic, Signal and Image Proceing The following definition i well-known in the literature and i widely ued: Definition A function ζ : I R, defined on the interval I in R, i aid to be convex on I if ζρτ ρ)τ ) ρζτ ) ρ)ζτ ), ) for all τ, τ I and ρ Alo we ay that ζ i concave on I, if the inequality given in ) hold in the revere direction Correponding to the definition of convex function the following double inequality ha played a very important role in variou field of cience Key word and phrae Convex function, Hermite-Hadamard inequality, fractional integral, Hölder inequality Mathematic Subject Claification 6D5 Received: 87 Accepted:87 6

2 PARAMETERIZED HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS 7 Theorem Let ζ be a convex function Then for τ, τ I with τ < τ, ξ τ, τ we have: ) τ τ τ ζ ζξ)dξ ζτ ) ζτ ) ) τ τ τ The order of inequality in ) i revered if ζ i concave Thi inequality i the widely ued Hermite-Hadamard inequality, which give an etimate from both ide of the mean ie from above and below of the mean value of a convex function and enure the integrability of any convex function too For more information about the Hermite-Hadamard inequality, the intereted reader can ee, 4 8, 5, 3, 4 We recall the definition of Riemann-Liouville R-L) fractional integral which we will ue in further reult: Definition 9) Let ζ L τ, τ with τ The Riemann-Liouville R-L) fractional integral operator J η ζ and J η ζ of order η > are defined by: τ τ J η τ ξ ζξ) = ξ ρ) η ζρ)dρ, with ξ > τ Γη) τ and J η ζξ) = τ ρ ξ) η ζρ)dρ, τ Γη) ξ with ξ < τ Here, Γη) repreent the Gamma function given by: Here J ζξ) = J ζξ) = ζξ) τ τ Riemann integral Γη) = e u u η du When η =, the R-L fractional integral reduce to E Set firtly examined Otrowki type inequalitie involving R-L fractional integral Alo, Sarikaya et al tudied the fractional form of the inequality ) For other of-late application of fractional derivative and fractional integral, one can ee 3, 4, 8,, 3 9 The fractional form of Hermite-Hadamard inequality i given below We will deign new bound for the difference of rightmot term in thi inequality Theorem ) Let ζ : τ, τ R be a poitive function with τ < τ and ζ L τ, τ If ζ i convex function on τ, τ, then the following inequality for R-L fractional integral hold: ) τ τ Γη ) ζ τ τ ) η J η τ ζτ ) J η τ ζτ ) ζτ ) ζτ ) 3) Remark By replacing η = in 3), we get the inequality )

3 8 M ADIL KHAN AND TU KHAN The following reult i due to Sarikaya et al, which contain a differentiable convex function In thi reult the difference of rightmot term in inequality 3) wa bounded Theorem 3 ) Let ζ : I R R be a twice differentiable function on I the interior of I) uch that τ, τ I with τ < τ If ζ i convex function on τ, τ, then the following inequality for fractional integral hold: ζτ ) ζτ ) Γη ) τ τ ) η J η ζτ τ ) J η ζτ τ ) τ τ ) η ) η ) ζ τ ) ζ τ ) In 7, Yu ming chu et al have dicovered an integral identity involving R-L fractional integral, which i given below: Lemma Let ζ : I, ) R be a twice differentiable function on I, uch that τ, τ I with τ < τ If ζ L τ, τ, then for η >, θ τ, τ the following identity hold: ζ θ) θ τ ) η τ θ) η) η )ζτ )τ θ) η η )ζτ )θ τ ) η η )τ τ ) Γη ) J η ζθ) J η ζθ) = θ τ ) η τ τ ) τ τ η )τ τ ) τ θ) η η )τ τ ) ρ η) ζ ρτ ρ)θ)dρ ρ η) ζ ρτ ρ)θ)dρ 4) Here, in the current paper, we deduce ome parameterized inequalitie of Hermite-Hadamard type via R-L fractional integral ee Theorem,,3,5,4) The novelty of thee reult i that they contain an arbitrary parameter θ from the interval of definition of twice differentiable convex or concave function Further more when the parameter θ i replaced by the midpoint of the interval we get different bound for the trapezoidal formula Main reult Before giving our main reult we introduce ome notation for the ake of implification Let ζ : I, ) R be a twice differentiable function on I and τ, τ I with τ < τ If ζ L τ, τ ζ i integrable on τ, τ ), then for all θ τ, τ and η >, we define L ζ by: L ζ θ, η, τ, τ ) = ζ θ) θ τ ) η τ θ) η) η )ζτ )τ θ) η η )ζτ )θ τ ) η η )τ τ ) Γη ) J η ζθ) J η ζθ) τ τ ) τ τ

4 PARAMETERIZED HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS 9 For θ = τ τ, we have ) τ τ L ζ, η, τ, τ = and by putting η = in thi, we get ) τ τ η ζτ ) ζτ ) Γη ) J η τ τ ζ τ τ ) τ ) J η ζ τ ) τ τ L ζ,, τ, τ = ζτ ) ζτ ) τ τ τ τ τ τ ), ζθ)dθ, which i the difference between the two right mot term in ) or the celebrated trapezoidal formula term Now we find out our firt parameterized bound Theorem Let all the requiite of Lemma hold Additionally, if ζ i convex function on τ, τ, then we have: L ζ θ, η, τ, τ ) θ τ ) η ζ τ ) η4 η ζ θ) τ θ) η ζ τ ) η4 τ τ )η 3) η ζ θ) ) Proof Uing Lemma, well known triangle inequality and convexity of ζ, we have L ζ θ, η, τ, τ ) θ τ ) η ρ η ) ζ ρτ ρ)θ) dρ η )τ τ ) τ θ) η η )τ τ ) θ τ ) η η )τ τ ) τ θ) η η )τ τ ) = θ τ ) η η )τ τ ) τ θ) η η )τ τ ) = θ τ ) η The proof i completed ρ η ) ζ ρτ ρ)θ) dρ ρ η) ρ ζ τ ) ρ) ζ θ) dρ ρ η) ρ ζ τ ) ρ) ζ θ) dρ η η 3) η η 3) ζ τ ) η4 η ζ θ) ) η 4)η ) ζ τ ) η 3)η ) ) ζ τ ) η 4)η ) η 3)η ) ) ζ θ) ) ζ θ) τ θ) η ζ τ ) η4 η ζ θ) τ τ )η 3)

5 3 M ADIL KHAN AND TU KHAN Corollary Under the hypothei of Theorem, we get ) ) L τ τ τ τ η ζ τ ) ζ τ ) ζ, η, τ, τ ) η ) Proof If we put θ = τ τ in inequality ) and ue the convexity of ζ, we get the deired inequality The aociated verion for power of the econd derivative abolute value of the function are included in the following theorem Theorem Let all the requiite of Lemma hold Additionally, if ζ i convex function on τ, τ for > and r =, then we have the following inequality: L ζ θ, η, τ, τ ) ) M r θ τ ) η ζ τ ) ζ θ) ) η )τ τ ) τ θ) η ζ τ ) ζ θ) ), 3) where M = Γ r)γ η ) η )Γ r η ) Proof Uing Lemma, triangle and Hölder inequalitie, we have L ζ θ, η, τ, τ ) θ τ ) η ρ η ) ζ ρτ ρ)θ) dρ η )τ τ ) τ θ) η η )τ τ ) θ τ ) η η )τ τ ) τ θ) η η )τ τ ) Uing the convexity of ζ, we get ρ η ) ζ ρτ ρ)θ) dρ r ρ η) r dρ r ρ η) r dρ ζ ρτ ρ)θ) dρ ζ ρτ ρ)θ) dρ ζ ρτ ρ)θ) dρ = ρ ζ τ ) ρ) ζ θ) ) dρ ζ τ ) ζ θ)

6 PARAMETERIZED HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS 3 Similarly alo ζ ρτ ρ)θ) dρ ρ ζ τ ) ρ) ζ θ) ) dρ = ζ τ ) ζ θ) ρ η) r dρ = u η u) r du = η Combining all the above inequalitie, we have the concluion 3) Γ r)γ η ) η )Γ r η ) = M Corollary Under the hypothei of Theorem, we have ) L τ τ M ζ, η, τ, τ ) ) r τ τ η η ) ) ζ τ ) τ τ ) ζ ) ζ τ ) τ τ ) ζ ) M τ τ η ζ τ ) ζ τ ) r 4) η ) Proof In inequality 3), if we take θ = τ τ, we get the firt bound in 4) The econd bound in 4) can be obtained by uing the convexity of ζ and the fact that: µ µ x ν y ν ) ω x ω ν µ yν ω, ν= ν= ν= for ω and x i, y i, where i =,, µ A more general parameterized bound can be prolonged in the following Theorem Theorem 3 Let all the requiite of Lemma hold Additionally, if ζ i convex function on τ, τ for, then we have the following inequality: L ζ θ, η, τ, τ ) ) η η η )τ τ ) ) η θ τ ) η η 3) ζ τ ) η 4)η ) η 3)η ) ζ θ) ) η τ θ) η η 3) ζ τ ) η 4)η ) η 3)η ) ζ θ) 5)

7 3 M ADIL KHAN AND TU KHAN Proof Uing Lemma and Power mean-inequality, we have L ζ θ, η, τ, τ ) θ τ ) η η )τ τ ) τ θ) η η )τ τ ) θ τ ) η η )τ τ ) τ θ) η η )τ τ ) ρ η) ζ ρτ ρ)θ) dρ ρ η) ζ ρτ ρ)θ) dρ ρ η )dρ ρ η) ζ ρτ ρ)θ) dρ ρ η )dρ ρ η) ζ ρτ ρ)θ) dρ 6) ince ζ i convex function on τ, τ, o we have ρ η ) ζ ρτ ρ)θ) dρ = ρ ρ η ) ζ τ ) ρ) ρ η ) ζ θ) dρ η η 3) ζ τ ) η 4)η ) η 3)η ) ζ θ), 7) imilarly we can write ρ η ) ζ ρτ ρ)θ) dρ = ρ ρ η ) ζ τ ) ρ) ρ η ) ζ θ) dρ η η 3) ζ τ ) η 4)η ) η 3)η ) ζ θ) 8) Alo we have ρ η )dρ = ) η η Now uing 7) and 8) in 6), we get 5)

8 PARAMETERIZED HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS 33 Corollary 3 Under the hypothei of Theorem 3, we have ) L τ τ ζ, η, τ, τ ) η τ τ ) η η η ) η η 3) ζ τ ) η η 3) ζ τ ) η 4)η ) η 3)η ) ζ τ τ ) η ) ) η η 4 η 3) η 3) η 4)η ) η 3)η ) ζ τ τ ) ) τ τ ) ζ τ ) ζ τ ) η ) ) ) 9) Proof In inequality 5), if we take θ = τ τ, we get the firt bound in 9) The econd bound in 9) can be obtained by uing the fact that: µ µ µ x ν y ν ) ω x ω ν yν ω, ν= ν= ν= for ω and x i, y i, where i =,, µ and then convexity of ζ Remark By putting = in 9), we get the inequality ) Intead of convexity, uing concavity property of the function we get two different inequalitie, which are given below Theorem 4 Let all the requiite of Lemma hold Additionally, If ζ i concave on τ, τ for each >, then the following inequality hold: L ζ θ, η, τ, τ ) θ τ ) η ) ζ η)τ η4)θ η3) τ θ) η ) ζ η)τ η4)θ η3) η )τ τ ) ) for each θ in τ, τ Proof By power mean inequality, we have ρ ζ τ ) ρ) ζ τ ) ) ρ ζ τ ) ρ) ζ τ ) ζ ρτ ρ)τ ), by concavity of ζ ) and therefore ζ ρτ ρ)τ ) ρ ζ τ ) ρ) ζ τ ),

9 34 M ADIL KHAN AND TU KHAN thi how that ζ i alo concave Now uing Lemma, triangular inequality and then Jenen integral inequality in turn, we have L ζ θ, η, τ, τ ) θ τ ) η η )τ τ ) τ θ) η η )τ τ ) ρ η) ζ ρτ ρ)θ) dρ ρ η) ζ ρτ ρ)θ) dρ θ τ ) η ρ η )ρτ ρ)θ)dρ ρ η )dρ ζ η )τ τ ) ρ η )dρ τ θ) η ρ η )ρτ ρ)θ)dρ ρ η )dρ ζ η )τ τ ) ρ η )dρ = θ τ ) η ) ζ η)τ η4)θ η3) τ θ) η ) ζ η)τ η4)θ η3) η )τ τ ) Corollary 4 Under the hypothei of Theorem 4, we have the following: ) ) L τ τ τ τ η ) 3η 8)τ η 4)τ ζ, η, τ, τ η ) ζ 4η 3) ) 3η 8)τ η 4)τ ζ 4η 3) ) τ τ η ) τ τ η ζ ) Proof By chooing θ = τ τ in the inequality ), we get the firt bound in ) The econd bound in ) i obtained by uing the concavity of ζ If we ue the property of concavity of ζ in another way, we get an another general reult, which i given below Theorem 5 Let all the requiite of Lemma hold Additionally, if ζ i concave function on τ, τ for > uch that r = Then we have the following inequality: θ τ L ζ θ, η, τ, τ ) M ) η ) ζ θτ τ θ) η ) ζ θτ r ) η )τ τ ) where M = Γr)Γ η ) η)γr η )

10 PARAMETERIZED HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS 35 Proof Applying Lemma, triangle and Hölder inequalitie, we have L ζ θ, η, τ, τ ) θ τ ) η ρ η ) ζ ρτ ρ)θ) dρ η )τ τ ) τ θ) η η )τ τ ) θ τ ) η η )τ τ ) τ θ) η η )τ τ ) ρ η ) ζ ρτ ρ)θ) dρ r ρ η) r dρ r ρ η) r dρ ζ ρτ ρ)θ) dρ ζ ρτ ρ)θ) dρ Since, ζ i concave on τ, τ, we can ue the Jenen inequality to get: ζ ρτ ρ)θ) dρ ζ ρτ ρ)θ)dρ ) = θ τ ζ, Similarly, Alo, ) ζ ρτ ρ)θ) dρ θ τ ζ ρ η) r dρ = u) r u η du = η Γ r)γ η ) η )Γ r η ) = M putting the lat three inequalitie in the above inequality, we get the required inequality in ) Corollary 5 Let all the requiite of Theorem 5 hold Then we have: ) L τ τ ζ, η, τ, τ M r τ τ ) η ) ) 3τ τ η η ) ζ τ 3τ 4 ζ 4 where M r η M = τ τ ) η ) ζ τ τ 3) Γ r)γ η ) η )Γ r η ) Proof By chooing θ = τ τ in the inequality ), we get the firt bound in 3) The econd bound in 3) i obtained uing concavity of ζ

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12 PARAMETERIZED HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS 37 4 L Chun, Some new inequalitie of Hermite-Hadamard type for α, m ) α, m )-convex function on coordinate, Journal of Function Space, 44), Article ID 97595, 7 page Department of Mathematic, Univerity of Pehawar, Pehawar, Pakitan addre: adilwati@gmailcom addre: tahirullah348@gmailcom