Generalization of different type integral inequalities for s-convex functions via fractional integrals
aa r X i v : . [ m a t h . C A ] A p r GENERALIZATION OF DIFFERENT TYPE INTEGRALINEQUALITIES FOR s − CONVEX FUNCTIONS VIAFRACTIONAL INTEGRALS ˙IMDAT ˙IS¸CAN
Abstract.
In this paper, a general integral identity for twice differentiablefunctions is derived. By using of this identity, the author establish somenew estimates on Hermite-Hadamard type and Simpson type inequalities for s − convex via Riemann Liouville fractional integral. Introduction
Following inequalities are well known in the literature as Hermite-Hadamardinequality and Simpson inequality respectively:
Theorem 1.
Let f : I ⊆ R → R be a convex function defined on the interval I ofreal numbers and a, b ∈ I with a < b . The following double inequality holds (1.1) f (cid:18) a + b (cid:19) ≤ b − a b Z a f ( x ) dx ≤ f ( a ) + f ( b )2 . Theorem 2.
Let f : [ a, b ] → R be a four times continuously differentiable mappingon ( a, b ) and (cid:13)(cid:13) f (4) (cid:13)(cid:13) ∞ = sup x ∈ ( a,b ) (cid:12)(cid:12) f (4) ( x ) (cid:12)(cid:12) < ∞ . Then the following inequality holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) f ( a ) + f ( b )2 + 2 f (cid:18) a + b (cid:19)(cid:21) − b − a b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) f (4) (cid:13)(cid:13)(cid:13) ∞ ( b − a ) . In [4], Hudzik and Maligranda considered among others the class of functionswhich are s − convex in the second sense. Definition 1.
A function f : [0 , ∞ ) → R is said to be s − convex in the secondsense if f ( αx + βy ) ≤ α s f ( x ) + β s f ( y ) for all x, y ∈ [0 , ∞ ) , α, β ≥ with α + β = 1 and for some fixed s ∈ (0 , . Thisclass of s − convex functions in the second sense is usually denoted by K s . It can be easily seen that for s = 1, s − convexity reduces to ordinary convexityof functions defined on [0 , ∞ ).We give some necessary definitions and mathematical preliminaries of fractionalcalculus theory which are used throughout this paper. Mathematics Subject Classification.
Key words and phrases.
Hermite–Hadamard inequality, Riemann–Liouville fractional integral,Simpson type inequalities, s − convex function. Definition 2.
Let f ∈ L [ a, b ] . The Riemann-Liouville integrals J αa + f and J αb − f ofoder α > with a ≥ are defined by J αa + f ( x ) = 1Γ( α ) x Z a ( x − t ) α − f ( t ) dt, x > a and J αb − f ( x ) = 1Γ( α ) b Z x ( t − x ) α − f ( t ) dt, x < b respectively, where Γ( α ) is the Gamma function defined by Γ( α ) = ∞ Z e − t t α − dt and J a + f ( x ) = J b − f ( x ) = f ( x ) . In the case of α = 1, the fractional integral reduces to the classical integral.Properties concerning this operator can be found [3, 5, 9].In [11], Sarikaya et.al established the following theorem: Theorem 3.
Let f : I ⊂ R → R be a twice differentiable mapping on I ◦ such that f ′′ ∈ L [ a, b ] , where a, b ∈ I with a < b . If | f ′′ | q is convex mapping on [ a, b ] for q ≥ , then the following inequality holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) f ( a ) + 4 f (cid:18) a + b (cid:19) + f ( b ) (cid:21) − b − a ) b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) ((cid:18) | f ′′ ( a ) | q + 133 | f ′′ ( b ) | q (cid:19) q + (cid:18) | f ′′ ( b ) | q + 133 | f ′′ ( a ) | q (cid:19) q ) . (1.2)In [12], Bo-Yan Xi and Feng Qi establish some new integral inequalities ofHermite-Hadamard type for h-convex functions as follow: Theorem 4.
Let
I, J ⊂ R be intervals, (0 , ⊂ J, and h : J → [0 , ∞ ) . Let f : I ⊂ R → R be a twice differentiable mapping on I ◦ such that f ′′ ∈ L [ a, b ] for a, b ∈ I with a < b . If ≤ λ ≤ and | f ′′ | q is an h − convex mapping on [ a, b ] for q ≥ , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 − λ ) f (cid:18) a + b (cid:19) + λ (cid:18) f ( a ) + f ( b )2 (cid:19) − b − a ) b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a )
16 [ H ( λ )] − q (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q Z t | λ − t | h ( t ) dt + | f ′′ ( a ) | q Z t | λ − t | h (1 − t ) dt q + (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q Z t | λ − t | h ( t ) dt + | f ′′ ( b ) | q C Z t | λ − t | h (1 − t ) dt q , N GENERALIZATION OF DIFFERENT TYPE INEQUALITIES 3 where H ( λ ) = (cid:26) λ − λ +13 , ≤ λ ≤ λ − , < λ ≤ β ( x, y ) = Γ( x )Γ( y )Γ( x + y ) = Z t x − (1 − t ) y − dt, x, y > , (2) The incomplete Beta function: β ( a, x, y ) = a Z t x − (1 − t ) y − dt, < a < , x, y > , (3) The hypergeometric function: F ( a, b ; c ; z ) = 1 β ( b, c − b ) Z t b − (1 − t ) c − b − (1 − zt ) − a dt, c > b > , | z | < h ( t ) = t s , s ∈ (0 , , we have the following inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 − λ ) f (cid:18) a + b (cid:19) + λ (cid:18) f ( a ) + f ( b )2 (cid:19) − b − a ) b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a )
16 [ H ( λ )] − q ((cid:18)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q H ( λ, s ) + | f ′′ ( a ) | q H ( λ, s ) (cid:19) q + (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q H ( λ, s ) + | f ′′ ( b ) | q H ( λ, s ) (cid:19) q ) , (1.3)where H ( λ, s ) = ( λ ) s +3 ( s +2)( s +3) − λs +2 + s +3 , ≤ λ ≤ λs +2 − s +3 , < λ ≤ , and H ( λ, s ) = (cid:20) λβ (2 , s + 1) − β (3 , s + 1)+4 λβ (2 λ ; 2 , s + 1) − β (2 λ ; 3 , s + 1) (cid:21) , ≤ λ ≤ λβ (2 , s + 1) − β (3 , s + 1) , < λ ≤ , In [7], Park establihed some new inequalities of the Simpson’s like and the Hermite-Hadamar-like type for s − convex functions as follows: Theorem 5.
Let f : I ⊂ [0 , ∞ ) → R be a twice differentiable function on I ◦ suchthat f ′′ is integrable on [ a, b ] , where a, b ∈ I ◦ with a < b . If | f ′′ | q is s − convex on [ a, b ] for some fixed s ∈ (0 , and q > , then for r ≥ the following inequalityholds: ˙IMDAT ˙IS¸CAN (a) b − a ) (cid:12)(cid:12) H ba ( f )( r ) (cid:12)(cid:12) (1.4) ≤ ((cid:20) r − r (cid:21) p +1 β (1 + p, p ) + 2 p +1 r p +1 ( p + 1) . F (cid:18) − p, p + 2; 2 r (cid:19)) p × (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( a ) | q s + 1 ! q + (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( b ) | q s + 1 ! q , where H ba ( f )( r )= (cid:18) − r (cid:19) (cid:26) f ( a ) + f ( b )2 (cid:27) + (cid:18)
12 + 1 r (cid:19) f (cid:18) a + b (cid:19) − b − a b Z a f ( x ) dx. (b) b − a ) (cid:12)(cid:12) R ba ( f )( r ) (cid:12)(cid:12) ≤ (cid:18) r (cid:19) . F (cid:0) − p, p + 2; − r (cid:1) ( p + 1) ! p × (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( a ) | q s + 1 ! q + (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( b ) | q s + 1 ! q , where R ba ( f )( r )= (cid:18)
12 + 1 r (cid:19) (cid:26) f ( a ) + f ( b )2 (cid:27) + (cid:18) − r (cid:19) f (cid:18) a + b (cid:19) − b − a b Z a f ( x ) dx. In recent years, many athors have studied errors estimations for Hermite-Hadamard’sinequality and Simpson’s on the class of s − convex mappings in the second senseand ( α, m ) − convex mappings; for refinements, counterparts, generalizations andnew Simpson’s type inequalities, see [2, 6, 7, 8, 10, 11, 12].The main aim of this article is to establish new generalization of Hermite Hadamard-type and Simpson-type inequalities for functions whose absolute values of secondderivatives are s − convex. To begin with, the author will derive a general integralidentity for twice differentiable mappings. By using this integral equality, the authorestablish some new inequalities of the Simpson-like and the Hermite-Hadamard-liketype for these functions. N GENERALIZATION OF DIFFERENT TYPE INEQUALITIES 5 Main Results
Let f : I ⊆ R → R be a differentiable function on I ◦ , the interior of I , throughoutthis section we will take I f ( x, λ, α, a, b )= (1 − λ ) (cid:20) ( x − a ) α + ( b − x ) α b − a (cid:21) f ( x ) + λ (cid:20) ( x − a ) α f ( a ) + ( b − x ) α f ( b ) b − a (cid:21) + (cid:18) α + 1 − λ (cid:19) " ( b − x ) α +1 − ( x − a ) α +1 b − a f ′ ( x ) − Γ ( α + 1) b − a [ J αx − f ( a ) + J αx + f ( b )]where a, b ∈ I with a < b , x ∈ [ a, b ] , λ ∈ [0 , α > Lemma 1.
Let f : I ⊆ R → R be a twice differentiable function on I ◦ such that f ′′ ∈ L [ a, b ] , where a, b ∈ I with a < b . Then for all x ∈ [ a, b ] , λ ∈ [0 , and α > we have: I f ( x, λ, α, a, b ) = ( x − a ) α +2 ( α + 1) ( b − a ) Z t (( α + 1) λ − t α ) f ′′ ( tx + (1 − t ) a ) dt (2.1) + ( b − x ) α +2 ( α + 1) ( b − a ) Z t (( α + 1) λ − t α ) f ′′ ( tx + (1 − t ) b ) dt. Proof.
By integration by parts twice and changing the variable, for x = a, we canstate Z t (( α + 1) λ − t α ) f ′′ ( tx + (1 − t ) a ) dt (2.2)= t (( α + 1) λ − t α ) f ′ ( tx + (1 − t ) a ) x − a (cid:12)(cid:12)(cid:12)(cid:12) − α + 1 x − a Z ( λ − t α ) f ′ ( tx + (1 − t ) a ) dt = (( α + 1) λ − f ′ ( x ) x − a − α + 1 x − a ( λ − t α ) f ( tx + (1 − t ) a ) x − a (cid:12)(cid:12)(cid:12)(cid:12) + Z αt α − f ( tx + (1 − t ) a ) x − a dt = (( α + 1) λ − f ′ ( x ) x − a + α + 1( x − a ) (1 − λ ) f ( x ) + λf ( a ) − α ( x − a ) α x Z a ( u − a ) α − f ( u ) du = (( α + 1) λ − f ′ ( x ) x − a + α + 1( x − a ) (cid:20) (1 − λ ) f ( x ) + λf ( a ) − Γ ( α + 1)( x − a ) α J αx − f ( a ) (cid:21) . Similarly, for x = b, we get Z t (( α + 1) λ − t α ) f ′′ ( tx + (1 − t ) b ) dt (2.3)= − (( α + 1) λ − f ′ ( x ) b − x + α + 1( b − x ) (cid:20) (1 − λ ) f ( x ) + λf ( b ) − Γ ( α + 1)( b − x ) α J αx + f ( b ) (cid:21) ˙IMDAT ˙IS¸CAN . Multiplying both sides of (2.2) and (2.3) by ( x − a ) α +2 ( α +1)( b − a ) and ( b − x ) α +2 ( α +1)( b − a ) , respectively,and adding the resulting identities, we obtain the desired result.For x = a and x = b, the identities I f ( a, λ, α, a, b ) = ( b − a ) α +1 ( α + 1) Z t (( α + 1) λ − t α ) f ′′ ( ta + (1 − t ) b ) dt, and I f ( b, λ, α, a, b ) = ( b − a ) α +1 ( α + 1) Z t (( α + 1) λ − t α ) f ′′ ( tb + (1 − t ) a ) dt, can be proved by performing an integration by parts twice in the integrals from theright side and changing the variable. (cid:3) Theorem 6.
Let f : I ⊂ [0 , ∞ ) → R be twice differentiable function on I ◦ suchthat f ′′ ∈ L [ a, b ] , where a, b ∈ I ◦ with a < b . If | f ′′ | q is s − convex on [ a, b ] for somefixed s ∈ (0 , and q ≥ , then for x ∈ [ a, b ] , λ ∈ [0 , and α > the followinginequality for fractional integrals holds | I f ( x, λ, α, a, b ) |≤ C − q ( α, λ ) ( ( x − a ) α +2 ( α + 1) ( b − a ) (cid:0) | f ′′ ( x ) | q C ( α, λ, s ) + | f ′′ ( a ) | q C ( α, λ, s ) (cid:1) q (2.4) + ( b − x ) α +2 ( α + 1) ( b − a ) (cid:0) | f ′′ ( x ) | q C ( α, λ, s ) + | f ′′ ( b ) | q C ( α, λ, s ) (cid:1) q ) , where C ( α, λ ) = ( α [( α +1) λ ]
1+ 2 α α +2 − ( α +1) λ + α +2 , ≤ λ ≤ α +1( α +1) λ − α +2 , α +1 < λ ≤ ,C ( α, λ, s ) = α [( α +1) λ ] α + s +2 α ( s +2)( α + s +2) − ( α +1) λs +2 + α + s +2 , ≤ λ ≤ α +1( α +1) λs +2 − α + s +2 , α +1 < λ ≤ ,C ( α, λ, s ) = β ( α + 2 , s + 1) − ( α + 1) λβ (2 , s + 1)2 ( α + 1) λβ (cid:16) [( α + 1) λ ] α ; 2 , s + 1 (cid:17) − β (cid:16) [( α + 1) λ ] α ; α + 2 , s + 1 (cid:17) , ≤ λ ≤ α +1 ( α + 1) λβ (2 , s + 1) − β ( α + 2 , s + 1) , α +1 < λ ≤ . N GENERALIZATION OF DIFFERENT TYPE INEQUALITIES 7
Proof.
From Lemma 1, property of the modulus and using the power-mean inequal-ity we have | I f ( x, λ, α, a, b ) | ≤ ( x − a ) α +2 ( α + 1) ( b − a ) Z t | ( α + 1) λ − t α | | f ′′ ( tx + (1 − t ) a ) | dt + ( b − x ) α +2 ( α + 1) ( b − a ) Z t | ( α + 1) λ − t α | | f ′′ ( tx + (1 − t ) b ) | dt ≤ ( x − a ) α +2 ( α + 1) ( b − a ) Z t | ( α + 1) λ − t α | dt − q × Z t | ( α + 1) λ − t α | | f ′′ ( tx + (1 − t ) a ) | q dt q + ( b − x ) α +2 ( α + 1) ( b − a ) Z t | t ( α + 1) λ − t α | dt − q × Z t | ( α + 1) λ − t α | | f ′′ ( tx + (1 − t ) b ) | q dt q . (2.5)Since | f ′ | q is s − convex on [ a, b ] we get Z t | ( α + 1) λ − t α | | f ′′ ( tx + (1 − t ) a ) | q dt ≤ Z t | ( α + 1) λ − t α | (cid:0) t s | f ′′ ( x ) | q + (1 − t ) s | f ′′ ( a ) | q (cid:1) dt = | f ′′ ( x ) | q C ( α, λ, s ) + | f ′′ ( a ) | q C ( α, λ, s ) , (2.6) Z t | ( α + 1) λ − t α | | f ′′ ( tx + (1 − t ) b ) | q dt ≤ Z t | ( α + 1) λ − t α | (cid:0) t s | f ′′ ( x ) | q + (1 − t ) s | f ′′ ( b ) | q (cid:1) dt = | f ′′ ( x ) | q C ( α, λ, s ) + | f ′′ ( b ) | q C ( α, λ, s ) , (2.7) ˙IMDAT ˙IS¸CAN where we use the fact that C ( α, λ, s )= Z t | ( α + 1) λ − t α | (1 − t ) s dt = ( α + 1) λ [( α +1) λ ] α Z t (1 − t ) s dt − [( α +1) λ ] α Z t α +1 (1 − t ) s dt − ( α + 1) λ Z [( α +1) λ ] α t (1 − t ) s dt + Z [( α +1) λ ] α t α +1 (1 − t ) s dt , ≤ λ ≤ α +1 ( α + 1) λ Z t (1 − t ) s dt − Z t α +1 (1 − t ) s dt, α +1 < λ ≤ β ( α + 2 , s + 1) − ( α + 1) λβ (2 , s + 1)2 ( α + 1) λβ (cid:16) [( α + 1) λ ] α ; 2 , s + 1 (cid:17) − β (cid:16) [( α + 1) λ ] α ; α + 2 , s + 1 (cid:17) , ≤ λ ≤ α +1 ( α + 1) λβ (2 , s + 1) − β ( α + 2 , s + 1) , α +1 < λ ≤ ,C ( α, λ, s )= Z t | ( α + 1) λ − t α | t s dt = α [( α +1) λ ] α + s +2 α ( s +2)( α + s +2) − ( α +1) λs +2 + α + s +2 , ≤ λ ≤ α +1( α +1) λs +2 − α + s +2 , α +1 < λ ≤ Z t | ( α + 1) λ − t α | dt = ( α [( α +1) λ ]
1+ 2 α α +2 − ( α +1) λ + α +2 , ≤ λ ≤ α +1( α +1) λ − α +2 , α +1 < λ ≤ (cid:3) Corollary 1.
In Theorem 6,(a) If we choose s = 1 , then we get: | I f ( x, λ, α, a, b ) |≤ C − q ( α, λ ) ( ( x − a ) α +1 b − a (cid:0) | f ′′ ( x ) | q C ( α, λ,
1) + | f ′′ ( a ) | q C ( α, λ, (cid:1) q + ( b − x ) α +1 b − a (cid:0) | f ′′ ( x ) | q C ( α, λ,
1) + | f ′′ ( b ) | q C ( α, λ, (cid:1) q ) . N GENERALIZATION OF DIFFERENT TYPE INEQUALITIES 9 (b) If we choose q = 1 , then we get: | I f ( x, λ, α, a, b ) | ≤ ( ( x − a ) α +1 b − a ( | f ′′ ( x ) | C ( α, λ, s ) + | f ′′ ( a ) | C ( α, λ, s ))+ ( b − x ) α +1 b − a ( | f ′′ ( x ) | C ( α, λ, s ) + | f ′′ ( b ) | C ( α, λ, s )) ) . (c) If we choose x = a + b , then we get: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − ( b − a ) α − I f (cid:18) a + b , λ, α, a, b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (1 − λ ) f (cid:18) a + b (cid:19) + λ (cid:18) f ( a ) + f ( b )2 (cid:19) − Γ ( α + 1) 2 α − ( b − a ) α (cid:20) J α ( a + b ) − f ( a ) + J α ( a + b ) + f ( b ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) α + 1) C − q ( α, λ ) ((cid:18)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q C ( α, λ, s ) + | f ′′ ( a ) | q C ( α, λ, s ) (cid:19) q + (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q C ( α, λ, s ) + | f ′′ ( b ) | q C ( α, λ, s ) (cid:19) q ) . (2.9) (d) If we choose x = a + b and λ = , then we get: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − ( b − a ) α − I f (cid:18) a + b , , α, a, b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) f ( a ) + 4 f (cid:18) a + b (cid:19) + f ( b ) (cid:21) − Γ ( α + 1) 2 α − ( b − a ) α (cid:20) J α ( a + b ) − f ( a ) + J α ( a + b ) + f ( b ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) α + 1) C − q (cid:18) α, (cid:19) ((cid:18)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q C (cid:18) α, , s (cid:19) + | f ′′ ( a ) | q C (cid:18) α, , s (cid:19)(cid:19) q + (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q C (cid:18) α, , s (cid:19) + | f ′′ ( b ) | q C (cid:18) α, , s (cid:19)(cid:19) q ) . (e) If we choose x = a + b , λ = , and α = 1 , then we get: (cid:12)(cid:12)(cid:12)(cid:12) I f (cid:18) a + b , , , a, b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) f ( a ) + 4 f (cid:18) a + b (cid:19) + f ( b ) (cid:21) − b − a ) b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) (cid:18) (cid:19) q ((cid:18)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q C (cid:18) , , s (cid:19) + | f ′′ ( a ) | q C (cid:18) , , s (cid:19)(cid:19) q + (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q C (cid:18) , , s (cid:19) + | f ′′ ( b ) | q C (cid:18) , , s (cid:19)(cid:19) q ) . (f ) If we choose x = a + b and λ = 0 , then we get: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − ( b − a ) α − I f (cid:18) a + b , , α, a, b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) a + b (cid:19) − Γ ( α + 1) 2 α − ( b − a ) α (cid:20) J α ( a + b ) − f ( a ) + J α ( a + b ) + f ( b ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) α + 1) (cid:18) α + 2 (cid:19) − q " (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q α + s + 2 + | f ′′ ( a ) | q β ( α + 2 , s + 1) q + " (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q α + s + 2 + | f ′′ ( b ) | q β ( α + 2 , s + 1) q . (g) If we choose x = a + b , λ = 0 , and α = 1 , then we get: (cid:12)(cid:12)(cid:12)(cid:12) I f (cid:18) a + b , , , a, b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) a + b (cid:19) − b − a ) b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) (cid:18) (cid:19) − q " (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q s + 3 + | f ′′ ( a ) | q β (3 , s + 1) q + " (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q s + 3 + | f ′′ ( b ) | q β (3 , s + 1) q . (h) If we choose x = a + b and λ = 1 , then we get: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − ( b − a ) α − I f (cid:18) a + b , , α, a, b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( b )2 − Γ ( α + 1) 2 α − ( b − a ) α (cid:20) J α ( a + b ) − f ( a ) + J α ( a + b ) + f ( b ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) α + 1) (cid:18) α ( α + 3)2 ( α + 2) (cid:19) − q × " α ( α + s + 3) (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q ( s + 2) ( α + s + 2) + | f ′′ ( a ) | q (( α + 1) β (2 , s + 1) − β ( α + 2 , s + 1)) q + " α ( α + s + 3) (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q ( s + 2) ( α + s + 2) + | f ′′ ( b ) | q (( α + 1) β (2 , s + 1) − β ( α + 2 , s + 1)) q . N GENERALIZATION OF DIFFERENT TYPE INEQUALITIES 11 (i) If we choose x = a + b , λ = 1 and α = 1 , then we get: (cid:12)(cid:12)(cid:12)(cid:12) I f (cid:18) a + b , , , a, b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( b )2 − b − a ) b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) (cid:18) (cid:19) − q × " ( s + 4) (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q ( s + 2) ( s + 3) + | f ′′ ( a ) | q (2 β (2 , s + 1) − β (3 , s + 1)) q + " ( s + 4) (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q ( s + 2) ( s + 3) + | f ′′ ( b ) | q (2 β (2 , s + 1) − β (3 , s + 1)) q . Remark 1.
In (c) of Corollary 1, if we choose α = 1 , then the inequality (2.9)reduces to the inequality (1.3). Remark 2.
In (e) of Corollary 1, if we choose s = 1 , we have the following Simpsontype inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) f ( a ) + 4 f (cid:18) a + b (cid:19) + f ( b ) (cid:21) − b − a ) b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) (cid:18) (cid:19) − q ((cid:18) (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q + 27972 | f ′′ ( a ) | q (cid:19) q + (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (cid:18) a + b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) q + 27972 | f ′′ ( b ) | q (cid:19) q ) . which is better than the inequality (1.2). Theorem 7.
Let f : I ⊂ [0 , ∞ ) → R be twice differentiable function on I ◦ suchthat f ′′ ∈ L [ a, b ] , where a, b ∈ I ◦ with a < b . If | f ′′ | q is s − convex on [ a, b ] for somefixed s ∈ (0 , and q > , then for x ∈ [ a, b ] , λ ∈ [0 , and α > the followinginequality for fractional integrals holds | I f ( x, λ, α, a, b ) | (2.10) ≤ C p ( α, λ, p ) ( ( x − a ) α +2 ( α + 1) ( b − a ) (cid:18) | f ′′ ( x ) | q + | f ′′ ( a ) | q s + 1 (cid:19) q + ( b − x ) α +2 ( α + 1) ( b − a ) (cid:18) | f ′′ ( x ) | q + | f ′′ ( b ) | q s + 1 (cid:19) q ) , where p = qq − and C ( α, λ, p )= p ( α +1)+1 , λ = 0 [( α +1) λ ] α +1) pα α β (cid:0) pα , p (cid:1) + [1 − ( α +1) λ ] p +1 α ( p +1) . F (cid:0) − pα , p + 2; 1 − ( α + 1) λ (cid:1) , < λ ≤ α +1[( α +1) λ ] p ( α +1)+1 α α β (cid:16) α +1) λ ; pα , p (cid:17) , α +1 < λ ≤ . Proof.
From Lemma 1, property of the modulus and using the H¨older inequalitywe have | I f ( x, λ, α, a, b ) | ≤ ( x − a ) α +2 ( α + 1) ( b − a ) Z t | ( α + 1) λ − t α | | f ′′ ( tx + (1 − t ) a ) | dt + ( b − x ) α +2 ( α + 1) ( b − a ) Z t | ( α + 1) λ − t α | | f ′′ ( tx + (1 − t ) b ) | dt ≤ ( x − a ) α +2 ( α + 1) ( b − a ) Z t p | ( α + 1) λ − t α | p dt p × Z | f ′′ ( tx + (1 − t ) a ) | q dt q + ( b − x ) α +2 ( α + 1) ( b − a ) Z t p | ( α + 1) λ − t α | p dt p × Z | f ′′ ( tx + (1 − t ) b ) | q dt q (2.11)Since | f ′ | q is s − convex on [ a, b ] we get Z | f ′′ ( tx + (1 − t ) a ) | q dt ≤ Z (cid:0) t s | f ′′ ( x ) | q + (1 − t ) s | f ′′ ( a ) | q (cid:1) dt = | f ′′ ( x ) | q + | f ′′ ( a ) | q s + 1 , (2.12) Z | f ′′ ( tx + (1 − t ) b ) | q dt ≤ Z (cid:0) t s | f ′′ ( x ) | q + (1 − t ) s | f ′′ ( b ) | q (cid:1) dt = | f ′′ ( x ) | q + | f ′′ ( b ) | q s + 1 , (2.13) N GENERALIZATION OF DIFFERENT TYPE INEQUALITIES 13 and Z t p | ( α + 1) λ − t α | p dt (2.14)= Z t ( α +1) p dt λ = 0 [( α +1) λ ] α Z t p [( α + 1) λ − t α ] p dt + Z [( α +1) λ ] α t p [ t α − ( α + 1) λ ] p dt, < λ ≤ α +11 Z t p [( α + 1) λ − t α ] p dt, α +1 < λ ≤ p ( α +1)+1 , λ = 0 [( α +1) λ ] α +1) pα α β (cid:0) pα , p (cid:1) + [1 − ( α +1) λ ] p +1 α ( p +1) . F (cid:0) − pα , p + 2; 1 − ( α + 1) λ (cid:1) , < λ ≤ α +1[( α +1) λ ] α +1) pα α β (cid:16) α +1) λ ; pα , p (cid:17) , α +1 < λ ≤ (cid:3) Corollary 2.
In Theorem 7(a) If we choose s = 1 , ten we get: | I f ( x, λ, α, a, b ) |≤ C p ( α, λ, p ) ( ( x − a ) α +2 ( α + 1) ( b − a ) (cid:18) | f ′′ ( x ) | q + | f ′′ ( a ) | q (cid:19) q + ( b − x ) α +2 ( α + 1) ( b − a ) (cid:18) | f ′′ ( x ) | q + | f ′′ ( b ) | q (cid:19) q ) . (b) If we choose x = a + b , then we get: | I f ( x, λ, α, a, b ) |≤ C p ( α, λ, p ) ( b − a ) α +1 ( α + 1) 2 α +2 (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( a ) | q s + 1 ! q + (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( b ) | q s + 1 ! q . (c) If we choose x = a + b , λ = , then we get: (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) f ( a ) + 4 f (cid:18) a + b (cid:19) + f ( b ) (cid:21) − Γ ( α + 1) 2 α − ( b − a ) α (cid:20) J α ( a + b ) − f ( a ) + J α ( a + b ) + f ( b ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C p (cid:18) α, , p (cid:19) ( b − a ) α + 1) (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( a ) | q s + 1 ! q + (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( b ) | q s + 1 ! q . (d) If we choose x = a + b , λ = , and α = 1 , then we get: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) f ( a ) + 4 f (cid:18) a + b (cid:19) + f ( b ) (cid:21) − b − a ) b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) C p (cid:18) , , p (cid:19) (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( a ) | q s + 1 ! q + (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( b ) | q s + 1 ! q , where C (cid:18) , , p (cid:19) = (cid:18) (cid:19) p β (1 + p, p ) + (cid:18) (cid:19) p . F (cid:18) − p, p + 2; 13 (cid:19) . (e) If we choose x = a + b and λ = 0 , then we get: (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) a + b (cid:19) − Γ ( α + 1) 2 α − ( b − a ) α (cid:20) J α ( a + b ) − f ( a ) + J α ( a + b ) + f ( b ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) (cid:18) p ( α + 1) + 1 (cid:19) p (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( a ) | q s + 1 ! q + (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( b ) | q s + 1 ! q . (f ) If we choose x = a + b and λ = 1 , then we get: (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( b )2 − Γ ( α + 1) 2 α − ( b − a ) α (cid:20) J α ( a + b ) − f ( a ) + J α ( a + b ) + f ( b ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) C p ( α, , p ) (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( a ) | q s + 1 ! q + (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( b ) | q s + 1 ! q , N GENERALIZATION OF DIFFERENT TYPE INEQUALITIES 15 where C ( α, , p ) = (1 + α ) p ( α +1)+1 α α β (cid:18)
11 + α ; 1 + pα , p (cid:19) . (g) If we choose x = a + b , λ = 1 and α = 1 , then we get: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( b )2 − b − a ) b Z a f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) (cid:18) β (cid:18)
12 ; 1 + p, p (cid:19)(cid:19) p (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( a ) | q s + 1 ! q + (cid:12)(cid:12) f ′′ (cid:0) a + b (cid:1)(cid:12)(cid:12) q + | f ′′ ( b ) | q s + 1 ! q . Remark 3.
In (b) of Corollary 2, if we take λ = − r , r ≥ , and α = 1 , thenthe inequality (2.10) reduces to the inequality (1.4). References [1] M. Abramowitz, I.A. Stegun, eds., Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tables, Dover, New York (1965).[2] M.I.Bhatti, M.Iqbal and S.S.Dragomir, Some new fractional integral Hermite-Hadamard typeinequalities, RGMIA Res. Rep. Coll., 16 (2013), Article 2.[3] R. Gorenflo, F. Mainardi, Fractional calculus; integral and differential equations of fractionalorder, Springer Verlag, Wien (1997), 223-276.[4] H. Hudzik and L. Maligranda, Some remarks on s − convex functions, Aequationes Math, 48(1994), 100-111.[5] S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional DifferentialEquations, John Wiley & Sons, USA (1993).[6] M. E. ¨Ozdemir, M. Avci and H. Kavurmaci, Hermite–Hadamard-type inequalities via( α, m ) − convexity, Computers and Mathematics with Applications 61 (2011), 2614–2620.[7] J. Park, Hermite and Simpson-like type inequalities for functions whose second derivatives inabsolute values at certain powera are s − convex, International Journal of Pure and AppliedMathematics, Vol. 78, No. 5, (2012), 587-604.[8] J. Park,On the Simpson-like type inequalities for twice differentiable ( α, m ) − convex mapping,International Journal of Pure and Applied Mathematics, Vol. 78, No. 5, (2012), 617–634.[9] I. Podlubni, Fractional Differential Equations, Academic Press, San Diego, 1999.[10] M.Z. Sarikaya and N. Aktan, On the generalization of some integral inequalities and theirapplications, Mathematical and Computer Modelling 54 (2011), 2175–2182.[11] M.Z. Sarikaya, E. Set and M.E. Ozdemir, On new inequalities of Simpson’s type for convexfunctions, RGMIA Res. Rep. Coll., 13, No. 2 (2010), Article 2.[12] B.-Y. Xi, F. Qi, Some inequalities of Hermite-Hadamard typr for h − convex functions, Ad-vances in Inequalities and Applications, 2 (2013), No.1, 1-15. Department of Mathematics, Faculty of Sciences and Arts, Giresun University, Gire-sun, Turkey
E-mail address ::