Ostrowski type Inequalities for m- and (alpha,m)-geometrically convex functions via Riemann-Louville Fractional integrals
aa r X i v : . [ m a t h . C A ] N ov OSTROWSKI TYPE INEQUALITIES FOR m − AND ( α, m ) − GEOMETRICALLY CONVEX FUNCTIONS VIARIEMANN-LOUVILLE FRACTIONAL INTEGRALS
MEVL ¨UT TUNC¸
Abstract.
In this paper, some new inequalities of Ostrowski type establishedfor the class of m − and ( α, m ) − geometrically convex functions which aregeneralizations of geometric convex functions. Introduction
The following result is known in the literature as Ostrowski’s inequality [1].
Theorem 1.
Let f : [ a, b ] → R be a differentiable mapping on ( a, b ) with theproperty that | f ′ ( u ) | ≤ M for all u ∈ ( a, b ) . Then the following inequality holds: (1.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − b − a Z ba f ( u ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M ( b − a )
14 + x − a + b b − a ! for all x ∈ [ a, b ] . The constant / is best possible in the sense that it cannot bereplaced by a smaller constant. This inequality gives an upper bound for the approximation of the integral av-erage b − a R ba f ( u ) du by the value f ( x ) at point x ∈ [ a, b ]. For recent results andgeneralizations concerning Ostrowski’s inequality, see [5]-[10] and the referencestherein.The following notations is well known in the literature. Definition 1.
A function f : I → R , ∅ 6 = I ⊆ R , where I is a convex set, is saidto be convex on I if inequality f ( tx + (1 − t ) y ) ≤ tf ( x ) + (1 − t ) f ( y ) holds for all x, y ∈ I and t ∈ [0 , . In particular in [3], Toader introduced the class of m − convex functions as ageneralizations of convexity as the following: Definition 2.
The function f : [0 , b ] → R is said to be m − convex, where m ∈ [0 , ,if for every x, y ∈ [0 , b ] and t ∈ [0 , we have (1.2) f ( tx + m (1 − t ) y ) ≤ tf ( x ) + m (1 − t ) f ( y ) Mathematics Subject Classification.
Key words and phrases.
Ostrowski’s inequality, m - and ( α, m )-geometrically convex functions. Moreover, in [2], Mihe¸san introduced the class of ( α, m ) − convex functions asthe following: Definition 3.
The function f : [0 , b ] → R is said to be ( α, m ) − convex, where ( α, m ) ∈ [0 , , if for every x, y ∈ [0 , b ] and t ∈ [0 , we have (1.3) f ( tx + m (1 − t ) y ) ≤ t α f ( x ) + m (1 − t α ) f ( y ) . In [4], Xi et al . introduced the class of m − and ( α, m ) − geometrically convexfunctions as the following: Definition 4. [4]
Let f ( x ) be a positive function on [0 , b ] and m ∈ (0 , . If (1.4) f (cid:16) x t y m (1 − t ) (cid:17) ≤ [ f ( x )] t [ f ( y )] m (1 − t ) holds for all x, y ∈ [0 , b ] and t ∈ [0 , , then we say that the function f ( x ) is m − geometrically convex on [0 , b ] . Obviously, if we set m = 1 in Definition 4, then f is just the ordinary geometri-cally convex on [0 , b ]. Definition 5. [4]
Let f ( x ) be a positive function on [0 , b ] and ( α, m ) ∈ (0 , × (0 , . If (1.5) f (cid:16) x t y m (1 − t ) (cid:17) ≤ [ f ( x )] t α [ f ( y )] m (1 − t α ) holds for all x, y ∈ [0 , b ] and t ∈ [0 , , then we say that the function f ( x ) is ( α, m ) − geometrically convex on [0 , b ] . Clearly, when we choose α = 1 in Definition 5, then f becomes the m − geometricallyconvex function on [0 , b ]. A very useful inequality will be given as following: Lemma 1. [4]
For x, y ∈ [0 , ∞ ) and m, t ∈ (0 , , if x < y and y ≥ , then x t y m (1 − t ) ≤ tx + (1 − t ) y. We give some necessary definitions and mathematical preliminaries of fractionalcalculus theory which are used throughout this paper.
Definition 6.
Let f ∈ L [ a, b ] . The Riemann-Liouville integrals J µa + f and J µb − f oforder µ > with a ≥ are defined by J µa + f ( x ) = 1Γ( µ ) x Z a ( x − t ) µ − f ( t ) dt, x > a and J µb − f ( x ) = 1Γ( µ ) b Z x ( t − x ) µ − f ( t ) dt, x < b respectively where Γ( µ ) = ∞ R e − t u µ − du. Here is J a + f ( x ) = J b − f ( x ) = f ( x ) . STROWSKI TYPE INEQUALITIES 3
In the case of µ = 1, the fractional integral reduces to the classical integral. Sev-eral researchers have interested on this topic and several papers have been writtenconnected with fractional integral inequalities see [11], [12], [13], [14], [15], [16],[18] and [19].The aim of this study is to establish some Ostrowski type inequalities for the classof functions whose derivatives in absolute value are m − and ( α, m ) − geometricallyconvex functions via Riemann-Liouville fractional integrals.2. Ostrowski type inequalities for m − and ( α, m ) − geometricallyconvex functions In order to prove our main theorems, we need the following lemma that has beenobtained in [19]:
Lemma 2.
Let f : [ a, b ] → R be a differentiable mapping on ( a, b ) with a < b. If f ′ ∈ L [ a, b ] , then for all x ∈ [ a, b ] and µ > we have: ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3) = ( x − a ) µ +1 b − a Z t µ f ′ ( tx + (1 − t ) a ) dt + ( b − x ) µ +1 b − a Z t µ f ′ ( tx + (1 − t ) b ) dt where Γ( µ ) = ∞ R e − t u µ − du. Theorem 2.
Let I ⊃ [0 , ∞ ) be an open interval and f : I → (0 , ∞ ) is differ-entiable. If f ′ ∈ L [ a, b ] and | f ′ | is decreasing and ( α, m ) − geometrically convexon [min { , a } , b ] for a ∈ [0 , ∞ ) , b ≥ , and | f ′ ( x ) | ≤ M ≤ , and ( α, m ) ∈ (0 , × (0 , , then the following inequality for fractional integrals with µ > holds: (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) (2.1) ≤ " ( x − a ) µ +1 + ( b − x ) µ +1 b − a × K ( α, m, µ ; k ( α )) where k ( α ) = ( M m R t µ M tα (1 − m ) dt , M < µ +1 , M = 1 . MEVL¨UT TUNC¸
Proof.
By Lemma 2 and since | f ′ | is decreasing and ( α, m )-geometrically convexon [min { , a } , b ] , we have (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( x − a ) µ +1 b − a Z t µ | f ′ ( tx + (1 − t ) a ) | dt + ( b − x ) µ +1 b − a Z t µ | f ′ ( tx + (1 − t ) b ) | dt ≤ ( x − a ) µ +1 b − a Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t a m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) dt + ( b − x ) µ +1 b − a Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t b m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) dt ≤ ( x − a ) µ +1 b − a Z t µ | f ′ ( x ) | t α | f ′ ( a ) | m (1 − t α ) dt + ( b − x ) µ +1 b − a Z t µ | f ′ ( x ) | t α | f ′ ( b ) | m (1 − t α ) dt ≤ ( x − a ) µ +1 b − a Z t µ M m + t α (1 − m ) dt + ( b − x ) µ +1 b − a Z t µ M m + t α (1 − m ) dt = M m b − a Z t µ M t α (1 − m ) dt h ( x − a ) µ +1 + ( b − x ) µ +1 i . If 0 < λ ≤ ≤ ∂, < u, v ≤ , then(2.2) λ u v ≤ λ uv . When M ≤ , by (2.2), we get that (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) (2.3) ≤ M m b − a h ( x − a ) µ +1 + ( b − x ) µ +1 i Z t µ M t α (1 − m ) dt ≤ M m b − a h ( x − a ) µ +1 + ( b − x ) µ +1 i Z t µ M tα (1 − m ) dt. The proof is completed. (cid:3)
Corollary 1.
Let I ⊃ [0 , ∞ ) be an open interval and f : I → (0 , ∞ ) is dif-ferentiable. If f ′ ∈ L [ a, b ] and | f ′ | is decreasing and m -geometrically convex on [min { , a } , b ] for a ∈ [0 , ∞ ) , b ≥ , and | f ′ ( x ) | ≤ M ≤ , and m ∈ (0 , , then thefollowing inequality for fractional integrals with µ > holds: (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ " ( x − a ) µ +1 + ( b − x ) µ +1 b − a × K (1 , m, µ ; k (1)) where k (1) = ( µ +1 , M = 1 M m R t µ M (1 − m ) t dt , M = 1 .Proof. We take α = 1 in (2.1), we get the required result. (cid:3) The corresponding version for powers of the absolute value of the first derivativeis incorporated in the following result:
STROWSKI TYPE INEQUALITIES 5
Theorem 3.
Let I ⊃ [0 , ∞ ) be an open interval and f : I → (0 , ∞ ) is dif-ferentiable. If f ′ ∈ L [ a, b ] and | f ′ | q is decreasing and ( α, m ) -geometrically con-vex on [min { , a } , b ] for a ∈ [0 , ∞ ) , b ≥ , p, q > and | f ′ ( x ) | ≤ M < ,x ∈ [min { , a } , b ] and ( α, m ) ∈ (0 , × (0 , , then the following inequality forfractional integrals with µ > holds: (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) (2.4) ≤ M m (cid:18) pµ + 1 (cid:19) p (cid:18) M qα (1 − m ) − qα (1 − m ) ln M (cid:19) q " ( x − a ) µ +1 + ( b − x ) µ +1 b − a where p − + q − = 1 .Proof. By Lemma 2 and since | f ′ | q is decreasing, and using the famous H¨olderinequality, we have (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( x − a ) µ +1 b − a Z t µ | f ′ ( tx + (1 − t ) a ) | dt + ( b − x ) µ +1 b − a Z t µ | f ′ ( tx + (1 − t ) b ) | dt ≤ ( x − a ) µ +1 b − a (cid:18)Z t pµ dt (cid:19) p (cid:18)Z (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t a m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt (cid:19) q + ( b − x ) µ +1 b − a (cid:18)Z t pµ dt (cid:19) p (cid:18)Z (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t b m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt (cid:19) q Since | f ′ | q is ( α, m ) − geometrically convex on [min { , a } , b ] and | f ′ ( x ) | ≤ M < Z (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t a m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt ≤ Z | f ′ ( x ) | qt α | f ′ ( a ) | mq (1 − t α ) dt ≤ Z M qt α + mq (1 − t α ) dt ≤ M mq Z M qt α (1 − m ) dt ≤ M mq Z M qtα (1 − m ) dt = M mq M qα (1 − m ) − qα (1 − m ) ln M and Z (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t b m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt ≤ Z | f ′ ( x ) | qt α | f ′ ( b ) | mq (1 − t α ) dt ≤ M mq M qα (1 − m ) − qα (1 − m ) ln M and by simple computation Z t pµ dt = 1 pµ + 1 . MEVL¨UT TUNC¸
Hence, we have (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M m (cid:18) pµ + 1 (cid:19) p (cid:18) M qα (1 − m ) − qα (1 − m ) ln M (cid:19) q " ( x − a ) µ +1 + ( b − x ) µ +1 b − a which completes the proof. (cid:3) Corollary 2.
Let I ⊃ [0 , ∞ ) be an open interval and f : I → (0 , ∞ ) is dif-ferentiable. If f ′ ∈ L [ a, b ] and | f ′ | q is decreasing and m -geometrically convexon [min { , a } , b ] for a ∈ [0 , ∞ ) , b ≥ , p, q > and | f ′ ( x ) | ≤ M < , x ∈ [min { , a } , b ] and m ∈ (0 , , then the following inequality for fractional integralswith µ > holds: (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M m (cid:18) pµ + 1 (cid:19) p (cid:18) M q (1 − m ) − q (1 − m ) ln M (cid:19) q " ( x − a ) µ +1 + ( b − x ) µ +1 b − a where /p + 1 /q = 1 . Proof.
We take α = 1 in (2.4), we get the required result. (cid:3) A different approach leads to the following result.
Theorem 4.
Let I ⊃ [0 , ∞ ) be an open interval and f : I → (0 , ∞ ) is differen-tiable. If f ′ ∈ L [ a, b ] and | f ′ | q is decreasing and ( α, m ) -geometrically convex on [min { , a } , b ] for a ∈ [0 , ∞ ) , b ≥ , q ≥ and | f ′ ( x ) | ≤ M < , x ∈ [min { , a } , b ] and α ∈ (0 , , m ∈ (0 , , then the following inequality for fractional integrals with µ > holds: (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) (2.5) ≤ M m (cid:18) µ + 1 (cid:19) − q (cid:18)Z t µ M qtα (1 − m ) dt (cid:19) q " ( x − a ) µ +1 + ( b − x ) µ +1 b − a Proof.
By Lemma 2 and since | f ′ | q is decreasing, and using the power mean in-equality, we have (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( x − a ) µ +1 b − a Z t µ | f ′ ( tx + (1 − t ) a ) | dt + ( b − x ) µ +1 b − a Z t µ | f ′ ( tx + (1 − t ) b ) | dt ≤ ( x − a ) µ +1 b − a (cid:18)Z t µ dt (cid:19) − q (cid:18)Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t a m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt (cid:19) q + ( b − x ) µ +1 b − a (cid:18)Z t µ dt (cid:19) − q (cid:18)Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t b m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt (cid:19) q STROWSKI TYPE INEQUALITIES 7
Since | f ′ | q is ( α, m )-geometrically convex and | f ′ ( x ) | ≤ M < Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t a m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt ≤ Z t µ | f ′ ( x ) | qt α | f ′ ( a ) | mq (1 − t α ) dt ≤ Z t µ M qt α + mq (1 − t α ) dt ≤ M mq Z t µ M qt α (1 − m ) dt ≤ M mq Z t µ M qtα (1 − m ) dt and similarly Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t b m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt ≤ M mq Z t µ M qtα (1 − m ) dt Hence, we have (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M m (cid:18) µ + 1 (cid:19) − q (cid:18)Z t µ M qtα (1 − m ) dt (cid:19) q " ( x − a ) µ +1 + ( b − x ) µ +1 b − a which comletes the proof. (cid:3) Corollary 3.
Let I ⊃ [0 , ∞ ) be an open interval and f : I → (0 , ∞ ) is differ-entiable. If f ′ ∈ L [ a, b ] and | f ′ | q is decreasing and m -geometrically convex on [min { , a } , b ] for a ∈ [0 , ∞ ) , b ≥ , q ≥ and | f ′ ( x ) | ≤ M < , x ∈ [min { , a } , b ] and m ∈ (0 , , then the following inequality for fractional integrals with µ > holds: (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M m (cid:18) µ + 1 (cid:19) − q (cid:18)Z t µ M qt (1 − m ) dt (cid:19) q " ( x − a ) µ +1 + ( b − x ) µ +1 b − a Proof.
We take α = 1 in (2.5), we get the required result. (cid:3) Corollary 4.
Let I ⊃ [0 , ∞ ) be an open interval and f : I → (0 , ∞ ) is differ-entiable. If f ′ ∈ L [ a, b ] and | f ′ | q is decreasing and geometrically convex on [ a, b ] for a ∈ [0 , ∞ ) , b ≥ , q ≥ and | f ′ ( x ) | ≤ M < , x ∈ [ a, b ] , then the followinginequality for fractional integrals with µ > holds: (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) (2.6) ≤ M (cid:18) µ + 1 (cid:19) " ( x − a ) µ +1 + ( b − x ) µ +1 b − a then, the inequality in (2.6) is special version of Corollary 3 of [19] .Proof. If we take α = 1 and m → (cid:3) MEVL¨UT TUNC¸
Corollary 5.
In Theorem 4, if we choose µ = 1 , then (2.5) reduces inequality above (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − b − a Z ba f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M m q (cid:18) M qα (1 − m ) − M qα (1 − m ) (cid:18) − M qα (1 − m ) (cid:19)(cid:19) q " ( x − a ) + ( b − x ) b − a ) Corollary 6.
Let f, g, a, b, µ, q be as in Theorem 4, and u, v > with u + v = 1 . Then (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) (2.7) ≤ M m " ( x − a ) µ +1 + ( b − x ) µ +1 b − a × (cid:18) µ + 1 (cid:19) − q u µ + u + v (cid:18) M qα (1 − m ) v − (cid:19) qα (1 − m ) ln M q Proof.
By Lemma 2 and since | f ′ | q is decreasing, and using the power mean in-equality, we have (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( x − a ) µ +1 b − a (cid:18)Z t µ dt (cid:19) − q (cid:18)Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t a m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt (cid:19) q + ( b − x ) µ +1 b − a (cid:18)Z t µ dt (cid:19) − q (cid:18)Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t b m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt (cid:19) q Since | f ′ | q is ( α, m )-geometrically convex and | f ′ ( x ) | ≤ M < Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t a m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt ≤ M mq Z t µ M qtα (1 − m ) dt Z t µ (cid:12)(cid:12)(cid:12) f ′ (cid:16) x t b m (1 − t ) (cid:17)(cid:12)(cid:12)(cid:12) q dt ≤ M mq Z t µ M qtα (1 − m ) dt By using the well known inequality cd ≤ uc u + vd v , we get that Z t µ M qtα (1 − m ) dt ≤ Z (cid:18) ut µu + vM qtα (1 − m ) v (cid:19) dt = u µ u + 1 + v M qα (1 − m ) v − M qα (1 − m ) v = u µ + u + v (cid:18) M qα (1 − m ) v − (cid:19) qα (1 − m ) ln M STROWSKI TYPE INEQUALITIES 9
Hence, we have (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M m (cid:18) µ + 1 (cid:19) − q u µ + u + v (cid:18) M qα (1 − m ) v − (cid:19) qα (1 − m ) ln M q " ( x − a ) µ +1 + ( b − x ) µ +1 b − a which comletes the proof. (cid:3) Remark 1.
In 2.7, if we choose q = 1 , then (2.7) reduces inequality above (cid:12)(cid:12)(cid:12)(cid:12) ( x − a ) µ + ( b − x ) µ b − a f ( x ) − Γ ( µ + 1) b − a (cid:2) J µx − f ( a ) + J µx + f ( b ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M m " ( x − a ) µ +1 + ( b − x ) µ +1 b − a u µ + u + v (cid:18) M α (1 − m ) v − (cid:19) α (1 − m ) ln M References [1] A. M. Ostrowski, ¨Uber die Absolutabweichung einer differentienbaren Funktionen von ihrenIntegralmittelwert, Comment. Math. Hel, 10 (1938), 226–227.[2] V.G. Mihe¸san, A generalization of the convexity, Seminar on Functional Equations, Approx.and Convex., Cluj-Napoca, Romania, 1993.[3] G.H. Toader, On a generalization of the convexity, Mathematica, 30 (53), (1988) 83-87.[4] B.-Y. Xi, R.-F. Bai and F. Qi: Hermite-Hadamard type inequalities for the m − and( α, m ) − geometrically convex functions. Aequationes Math., doi: 10.1007/s00010-011-0114-x.[5] M. Alomari, M. Darus, Some Ostrowski type inequalities for convex functions with applica-tions, RGMIA 13 (1) (2010) article No. 3. Preprint.[6] M. Alomari, M. Darus, Some Ostrowski type inequalities for quasi-convex functions withapplications to special means, RGMIA 13 (2) (2010) article No. 3. Preprint.[7] M. Alomari, M. Darus, S.S. Dragomir, P. Cerone, Ostrowski type inequalities for functionswhose derivatives are s -convex in the second sense, Appl. Math. Lett. Volume 23 (2010)1071-1076.[8] P. Cerone, S.S. Dragomir, Ostrowski type inequalities for functions whose derivatives satisfycertain convexity assumptions, Demonstratio Math. 37 (2) (2004), 299-308.[9] N.S. Barnett, P. Cerone, S.S. Dragomir, M.R. Pinheiro, A. Sofo, Ostrowski type inequalitiesfor functions whose modulus of derivatives are convex and applications, RGMIA Res. Rep.Coll. 5 (2) (2002) Article 1. Online: http://rgmia.vu.edu.au/v5n2.html.[10] S.S. Dragomir, A. Sofo, Ostrowski type inequalities for functions whose derivatives are convex,in: Proceedings of the 4th International Conference on Modelling and Simulation, November11-13, 2002. Victoria University, Melbourne, Australia, RGMIA Res. Rep. Coll. 5 (2002)Supplement, Article 30. Online: http://rgmia.vu.edu.au/v5(E).html.[11] Belarbi, S. and Dahmani, Z.: On some new fractional integral inequalities, J. Ineq. Pure andAppl. Math., 10(3), Art. 86 (2009).[12] Dahmani, Z.: New inequalities in fractional integrals, International Journal of NonlinearScience, 9(4), 493-497 (2010).[13] Dahmani, Z.: On Minkowski and Hermite-Hadamard integral inequalities via fractional inte-gration, Ann. Funct. Anal. 1(1), 51-58 (2010).[14] Dahmani, Z., Tabharit, L. and Taf, S.: Some fractional integral inequalities, Nonl. Sci. Lett.A., 1(2), 155-160 (2010).[15] Dahmani, Z., Tabharit, L. and Taf, S.: New generalizations of Gr¨uss inequality usingRiemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2(3), 93-99 (2010). [16] ¨Ozdemir, M.E., Kavurmacı, H. and Avcı, M.: New inequalities of Ostrowski type for map-pings whose derivatives are ( α, m ) − convex via fractional integrals, RGMIA Research ReportCollection, 15, Article 10, 8 pp (2012).[17] ¨Ozdemir, M.E., Kavurmacı, H. and Yıldız, C¸ .: Fractional integral inequalities via s − convexfunctions, arXiv:1201.4915v1 [math.CA] 24 Jan 2012.[18] Sarıkaya, M. Z., Set, E., Yaldiz, H. and Ba¸sak, N.: Hermite-Hadamard’s inequalities for frac-tional integrals and related fractional inequalities, Mathematical and Computer Modelling,In Press.[19] Set, E.: New inequalities of Ostrowski type for mappings whose derivatives are s − convex inthe second sense via fractional integrals, Comput. Math. Appl., 63 (2012) 1147-1154. Kilis 7 Aralık University, Faculty of Science and Arts, Department of Mathematics,Kilis, 79000, Turkey.
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