Hermite--Hadamard type inequalities via differentiable h φ -preinvex functions for Fractional Integrals
aa r X i v : . [ m a t h . C A ] N ov HERMITE–HADAMARD TYPE INEQUALITIES VIADIFFERENTIABLE h ϕ − PREINVEX FUNCTIONS FORFRACTIONAL INTEGRALS
ABDULLAH AKKURT AND H ¨USEYIN YILDIRIM
Abstract.
In this paper, we consider a new class of convex functions whichis called h ϕ − preinvex functions. We prove several Hermite–Hadamard typeinequalities for differentiable h ϕ − preinvex functions via Fractional Integrals.Some special cases are also discussed. Our results extend and improve thecorresponding ones in the literature. Introduction
The following inequality is well-known in the literature as Hermite-Hadamardinequality. Let f : I ⊆ R → R be a convex function with a < b and a, b ∈ I . Thenthe following holds(1.1) f ( a + b ≤ b − a b Z a f ( x ) dx ≤ f ( a ) + f ( b )2 . Recently, Hermite-Hadamard type inequality has been the subject of intensiveresearch. Hermite–Hadamard inequality can be considered as necessary and suffi-cient condition for a function to be convex. It provides estimates of the mean valueof continuous convex function.In recent years, several extensions and generalizations have been proposed forclassical convexity (see [1 − ϕ − convex functions introduced by Noor [10]. Noor [10] hasinvestigated the basic properties of ϕ − convex functions and showed that ϕ − convexfunctions are nonconvex functions. Noor [9] extended Hermite–Hadamard typeinequalities for ϕ − convex functions.Motivated by the ongoing research on generalizations and extensions of classicalconvexity, Varosanec [23] introduced the class of h − convex functions. Noor etal. [13] introduced another generalization of classical convexity which is calledas h ϕ − convex functions. Finally Noor et al. Several Hermite–Hadamard typeinequalities are proved for h ϕ − preinvex functions[20].Weir and Mond [24] investigated the class of preinvex functions. It is well-knownthat the preinvex functions and invex sets may not be convex functions and convexsets. For recent investigation on preinvexity (see [1 , , − , , h ϕ − preinvex functions. This class includes Key words and phrases. integral inequalities, fractional integrals, Hermite-Hadamard Inequal-ity, Preinvex functions, H¨older’s inequality. several known and new classes of convex functions such as ϕ − convex functions [10],preinvex functions [21], h − convex functions [23] as special cases. We derive sev-eral new Hermite–Hadamard type fractional integral inequalities for h ϕ − preinvexfunctions and their variant forms. Results obtained in this paper continue to holdfor these special cases. Our results represent significant generalized of the previ-ous results. The interested readers are encouraged to find novel and innovativeapplications of h ϕ − preinvex functions and fractional integrals.2. Preliminares
Let R n be the finite dimensional Euclidian space. Also let 0 ≤ ϕ ≤ π Definition 1. ([12])
The set K ϕn in R n is said to be ϕ − invex at u with respect to ϕ ( . ) , if there exists a bifunction η ( ., . ) : K ϕn × K ϕn → R , such that (2.1) u + te iϕ η ( v, u ) ∈ K ϕn , ∀ u, v ∈ K ϕn , t ∈ [0 , . The ϕ − invex set K ϕn is also called ϕn -connected set. Note that the convex set with ϕ = 0 and η ( v, u ) = v − u is a ϕ − invex set, but the converse is not true. Definition 2. ([10])
Let K ϕ be a set in R n . Then the set K ϕ is said to be ϕ − convexwith respect to ϕ , if and only if u + te iϕ ( v, u ) ∈ K ϕ , ∀ u, v ∈ K ϕ , t ∈ [0 , . For ϕ = 0 , the set K ϕ reduces to the classical convex set K . That is, u + t ( v, u ) ∈ K, ∀ u, v ∈ K, t ∈ [0 , . Definition 3. ([24])
A set K n is said to be invex set with respect to bifunction η ( ., . ) , if u + tη ( v, u ) ∈ K n , ∀ u, v ∈ K n , t ∈ [0 , . The invex set K n is also called η − connected set. Definition 4. ([20])
Let h : J ⊆ R → R be a nonnegative function. A function f onthe set K ϕn is said to be h ϕ − preinvex function with respect to ϕ and bifunction η ,if (2.2) f ( u + te iϕ η ( v, u )) ≤ h (1 − t ) f ( u ) + h ( t ) f ( v ) , ∀ u, v ∈ K ϕn , t ∈ [0 , . Remark 1.
One can deduce several known concepts from Definition as: (1) For h ( t ) = t Definition 4 reduces to the definition for ϕ − preinvex functions(see [12]).(2) For ϕ = 0 Definition 4 reduces to the definition for h − preinvex functions(see [17]).(3) For η ( v, u ) = v − u Definition 4 reduces to the definition for h ϕ − convexfunctions (see [13]).(4) For ϕ = 0 and η ( v, u ) = v − u Definition 4 reduces to the definition for h − convex functions (see [23]).Now we discuss some special cases of Definition 4. I . For h ( t ) = t s where s ∈ (0 ,
1] in (2 .
2) we have the definition for s ϕ − preinvexfunctions. ERMITE–HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE ... Definition 5.
A function f on the set K ϕn is said to be s ϕ − preinvex function withrespect to ϕ and η , if (2.3) f ( u + te iϕ η ( v, u )) ≤ (1 − t ) s f ( u ) + t s f ( v ) , ∀ u, v ∈ K ϕn , t ∈ [0 , . II . For h ( t ) = 1 in (2 .
2) we have the definition for P ϕ − preinvex functions. Definition 6.
A function f on the set K ϕn is said to be s ϕ − preinvex function withrespect to ϕ and η , if (2.4) f ( u + te iϕ η ( v, u )) ≤ f ( u ) + f ( v ) , ∀ u, v ∈ K ϕn , t ∈ [0 , . Definition 7. ([25])
Let f ∈ L [ a, b ] . The Riemann-Liouville fractional integral J αa + f ( x ) and J αb − f ( x ) of order α > are defined by (2.5) J αa + [ f ( x )] = α ) x Z a ( x − t ) α − f ( t ) dt x > a and (2.6) J αb − [ f ( x )] = α ) b Z x ( t − x ) α − f ( t ) dt x < b respectively, where Γ ( α ) = ∞ Z e − u u α − du is Gamma function and J a + f ( x ) = J b − f ( x ) = f ( x ) . Main results
In this section, we will discuss our main results for h ϕ − preinvex functions andfractional integrals.Using the technique of [1] and [20], we prove the following Lemma which play akey role in our study. Lemma 1.
Let I ⊆ R be a open invex set with respect to bifunction η : I × I → R where η ( b, a ) > . If f ′ ∈ L [ a, a + e iϕ η ( b, a )] and α ≥ , then (3.1) (cid:2) f ( a ) + f ( a + e iϕ η ( b, a )) (cid:3) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o = e iϕ η ( b, a ) Z [ t α − (1 − t ) α ] f ′ ( a + te iϕ η ( b, a )) dt. ABDULLAH AKKURT AND H¨USEYIN YILDIRIM
Proof.
Let Z [(1 − t ) α − t α ] f ′ ( a + te iϕ η ( b, a )) dt = ((1 − t ) α − t α ) f ( a + te iϕ η ( b, a )) e iϕ η ( b, a ) (cid:12)(cid:12)(cid:12)(cid:12) + αe iϕ η ( b, a ) Z h (1 − t ) α − + t α − i f ( a + te iϕ η ( b, a )) dt = − (cid:2) f ( a ) + f ( a + e iϕ η ( b, a )) (cid:3) e iϕ η ( b, a ) + αe iϕ η ( b, a ) Z h (1 − t ) α − + t α − i f ( a + te iϕ η ( b, a )) dt = − e iϕ η ( b,a ) (cid:2)(cid:2) f ( a ) + f ( a + e iϕ η ( b, a )) (cid:3) − α a + e iϕ η ( b,a ) Z a (cid:20)(cid:16) − u − ae iϕ η ( b,a ) (cid:17) α − + (cid:16) u − ae iϕ η ( b,a ) (cid:17) α − (cid:21) f ( u ) due iϕ η ( b,a ) = − (cid:2) f ( a ) + f ( a + e iϕ η ( b, a )) (cid:3) e iϕ η ( b, a )+ αe iϕ η ( b, a ) a + e iϕ η ( b,a ) Z a (cid:20)(cid:0) a + e iϕ η ( b, a ) − u (cid:1) α − + (cid:16) u − ae iϕ η ( b,a ) (cid:17) α − (cid:21) f ( u ) du = − (cid:2) f ( a ) + f ( a + e iϕ η ( b, a )) (cid:3) e iϕ η ( b, a ) + Γ ( α + 1)[ e iϕ η ( b, a )] α +1 h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i . This complete the proof. (cid:3)
Theorem 1.
Let I ⊆ R be a open invex set with respect to bifunction η : I × I → R . Suppose f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . Let (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) be h ϕ − preinvex function. Then, for η ( b, a ) > and α > , (3.2) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ e iϕ η ( b, a ) h(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12)i Z | (1 − t ) α − t α | h ( t ) dt. Proof.
Using Lemma 2 and h ϕ − preinvexity of (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) , we have (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ e iϕ η ( b, a ) Z | (1 − t ) α − t α | (cid:12)(cid:12)(cid:12) f ′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) dt ≤ e iϕ η ( b, a ) Z | (1 − t ) α − t α | (cid:16) h (1 − t ) (cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) + h ( t ) (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12)(cid:17) dt ERMITE–HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE ... = e iϕ η ( b, a ) Z | (1 − t ) α − t α | h (1 − t ) (cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) dt + Z | (1 − t ) α − t α | h ( t ) (cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) dt = e iϕ η ( b, a ) Z | t α − (1 − t ) α | h ( t ) (cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) dt + Z | (1 − t ) α − t α | h ( t ) (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) dt = e iϕ η ( b, a ) h(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12)i Z | (1 − t ) α − t α | h ( t ) dt. This completes the proof. (cid:3)
Now we have some special cases. I. If h ( t ) = t , then we have the result for ϕ − preinvexity. Corollary 1.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) is ϕ − preinvex on I, then, for η ( b, a ) > , (3.3) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ e iϕ η ( b, a ) h(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12)i α + 2 (cid:20) − α +2 (cid:21) . II. If h ( t ) = t s , then we have the result for s ϕ − preinvexity Corollary 2.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) is s ϕ − preinvex on I , then, for η ( b, a ) > and s ∈ (0 , , (3.4) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ e iϕ η ( b, a ) h(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12)i h B ( α + 1 , s + 1) (cid:18) − B ( s + 1 , α + 1) + 1 α + s + 1 (cid:20) − s + α (cid:21)(cid:19)(cid:21) . III. If h ( t ) = 1, then we have the result for P ϕ − preinvexity. Corollary 3.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) is P ϕ − preinvex on I , then, for η ( b, a ) > , (3.5) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ e iϕ η ( b, a ) h(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12)i α + 1 (cid:20) − α (cid:21) . ABDULLAH AKKURT AND H¨USEYIN YILDIRIM
Theorem 2.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . Let (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) p be h ϕ − preinvex on I , p + q = 1 and η ( b, a ) > . Then (3.6) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) αp + 1 (cid:19) p e iϕ η ( b, a ) (cid:20) − αp (cid:21) p (cid:20)(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − (cid:21) Z h ( t ) dt p − p . Proof.
Using Lemma 2, we have (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e iϕ η ( b, a ) Z [(1 − t ) α − t α ] f ′ ( a + te iϕ η ( b, a )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e iϕ η ( b, a ) Z | (1 − t ) α − t α | (cid:12)(cid:12)(cid:12) f ′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) dt. From Holders inequality, we have (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ e iϕ η ( b, a ) Z | (1 − t ) α − t α | p dt p Z (cid:12)(cid:12)(cid:12) f ′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) q dt q = e iϕ η ( b, a ) Z ((1 − t ) α − t α ) p dt + Z ( t α − (1 − t ) α ) p dt p × Z (cid:12)(cid:12)(cid:12) f ′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) q dt q . Here h ϕ − preinvexity of (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) p , we have following inequality (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ p ( αp + 1) p e iϕ η ( b, a ) (cid:20) − αp (cid:21) p Z h h (1 − t ) (cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) q + h ( t ) (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) q i dt q = (cid:18) αp + 1 (cid:19) p e iϕ η ( b, a ) (cid:20) − αp (cid:21) p (cid:20)(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − (cid:21) Z h ( t ) dt p − p . This completes the proof. (cid:3)
Now we have some special cases for (3 . ERMITE–HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE ... I. If h ( t ) = t , then we have the result for ϕ − preinvexity. Corollary 4.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . Let (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) p be ϕ − preinvex on I, p + q = 1 and η ( b, a ) > . Then (3.7) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) αp + 1 (cid:19) p e iϕ η ( b, a ) (cid:20) − αp (cid:21) p (cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − p − p . II. If h ( t ) = t s , then we have the result for s ϕ − preinvexity Corollary 5.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . Let (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) p be s ϕ − preinvex on I , p + q = 1 , η ( b, a ) > and s ∈ (0 , . Then (3.8) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) αp + 1 (cid:19) p e iϕ η ( b, a ) (cid:20) − αp (cid:21) p (cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − s + 1 p − p . III. If h ( t ) = 1, then we have the result for P ϕ − preinvexity. Corollary 6.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . Let (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) p be P ϕ − preinvex on I , p + q = 1 and η ( b, a ) > . Then (3.9) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) αp + 1 (cid:19) p e iϕ η ( b, a ) (cid:20) − αp (cid:21) p (cid:26)(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − (cid:27) p − p . Theorem 3.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . Let (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) q , q > is h ϕ − preinvex on I , then, for η ( b, a ) > , (3.10) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) α + 1 (cid:19) p (cid:20) − α (cid:21) p e iϕ η ( b, a ) (cid:20)(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − (cid:21) Z | (1 − t ) α − t α | h ( t ) dt p − p . ABDULLAH AKKURT AND H¨USEYIN YILDIRIM
Proof.
Using Lemma 2, we have (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e iϕ η ( b, a ) Z [(1 − t ) α − t α ] f ′ ( a + te iϕ η ( b, a )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e iϕ η ( b, a ) Z | (1 − t ) α − t α | (cid:12)(cid:12)(cid:12) f ′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) dt Using power-mean inequality, we have (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ e iϕ η ( b, a ) Z | (1 − t ) α − t α | dt p Z (cid:12)(cid:12)(cid:12) f ′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) q dt q = e iϕ η ( b, a ) Z ((1 − t ) α − t α ) dt + Z ( t α − (1 − t ) α ) dt p × Z (cid:12)(cid:12)(cid:12) f ′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) q dt q now using h ϕ − preinvexity of (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) q , we have (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ p ( α + 1) p e iϕ η ( b, a ) (cid:20) − α (cid:21) p Z h h (1 − t ) (cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) q + h ( t ) (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) q i dt q = (cid:18) α + 1 (cid:19) p e iϕ η ( b, a ) (cid:20) − α (cid:21) p (cid:20)(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − (cid:21) Z | (1 − t ) α − t α | h ( t ) dt p − p This completes the proof. (cid:3)
Now we have some special cases for (3 . I. If h ( t ) = t , then we have the result for ϕ − preinvexity. Corollary 7.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) q , q > is ϕ − preinvex on I, then, for η ( b, a ) > , (3.11) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) α + 1 (cid:19) p (cid:20) − α (cid:21) p e iϕ η ( b, a ) (cid:26)(cid:20)(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − (cid:21) α + 2 (cid:20) − α +2 (cid:21)(cid:27) p − p . ERMITE–HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE ... II. If h ( t ) = t s , then we have the result for s ϕ − preinvexity Corollary 8.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) q , q > is s ϕ − preinvex on I , then, for η ( b, a ) > and s ∈ (0 , , (3.12) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) α +1 − α ( α + 1) (cid:19) e iϕ η ( b, a ) (cid:26)(cid:20)(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − (cid:21) x (cid:20) B ( α + 1 , s + 1) − B ( s + 1 , α + 1) + 1 α + s + 1 (cid:20) − s + α (cid:21)(cid:21)(cid:27) p − p . III. If h ( t ) = 1, then we have the result for P ϕ − preinvexity. Corollary 9.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Supposethat f : I → R is a differentiable function such that f ′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′ (cid:12)(cid:12)(cid:12) q , q > is P ϕ − preinvex on I , then, for η ( b, a ) > (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α h J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) i(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) α +1 − α ( α + 1) (cid:19) e iϕ η ( b, a ) (cid:26)(cid:12)(cid:12)(cid:12) f ′ ( a ) (cid:12)(cid:12)(cid:12) pp − + (cid:12)(cid:12)(cid:12) f ′ ( b ) (cid:12)(cid:12)(cid:12) pp − (cid:27) p − p . Using the technique of [1], we prove the following result which helps us in provingour next results.
Lemma 2.
Let I ⊆ R be a open invex set with respect to η : I × I → R where η ( b, a ) > . If f ′′ ∈ L [ a, a + e iϕ η ( b, a )] , then (3.14) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o = (cid:2) e iϕ η ( b, a ) (cid:3) Z " − (1 − t ) α +1 − t α +1 α + 1 f ′′ ( a + te iϕ η ( b, a )) dt. Proof.
Let Z " − (1 − t ) α +1 − t α +1 α + 1 f ′′ ( a + te iϕ η ( b, a )) dt = 1 − (1 − t ) α +1 − t α +1 α + 1 f ′ ( a + te iϕ η ( b, a )) e iϕ η ( b, a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − e iϕ η ( b, a ) Z [(1 − t ) α − t α ] f ′ ( a + te iϕ η ( b, a )) dt = − e iϕ η ( b, a ) Z [(1 − t ) α − t α ] f ′ ( a + te iϕ η ( b, a )) dt = 1( e iϕ η ( b, a )) (cid:20)(cid:2) f ( a ) + f ( a + e iϕ η ( b, a )) (cid:3) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:21) This completes the proof. (cid:3)
Theorem 4.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Let f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) , is h ϕ − preinvex on I , then, for η ( b, a ) > , (3.15) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) (cid:16)(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12)(cid:17) Z " − (1 − t ) α +1 − t α +1 α + 1 h ( t ) dt . Proof.
Using Lemma 3, we have (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) e iϕ η ( b, a ) (cid:3) Z " − (1 − t ) α +1 − t α +1 α + 1 f ′′ ( a + te iϕ η ( b, a )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) Z " − (1 − t ) α +1 − t α +1 α + 1 f ′′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) dt ≤ (cid:2) e iϕ η ( b, a ) (cid:3) Z " − (1 − t ) α +1 − t α +1 α + 1 h (1 − t ) (cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) + h ( t ) (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12)(cid:17) dt = (cid:2) e iϕ η ( b, a ) (cid:3) (cid:16)(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12)(cid:17) Z " − (1 − t ) α +1 − t α +1 α + 1 h ( t ) dt This completes the proof. (cid:3)
Now, we discuss some special cases for (3 . I. If h ( t ) = t , then we have the result for ϕ − preinvexity. Corollary 10.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Suppose that f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is ϕ − preinvex on I, then, for η ( b, a ) > , (3.16) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) (cid:26) α α + 1) ( α + 2) (cid:16)(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12)(cid:17)(cid:27) . II. If h ( t ) = t s , then we have the result for s ϕ − preinvexity Corollary 11.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Let f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is s ϕ − preinvex on I , then, for η ( b, a ) > and s ∈ (0 , , (3.17) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) ((cid:16)(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12)(cid:17) α + s + 2) ( s + 1) − B ( s + 1 , α ) α + 1 !) . ERMITE–HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE ... III. If h ( t ) = 1, then we have the result for P ϕ − preinvexity. Corollary 12.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Let f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is P ϕ − preinvex on I , then, for η ( b, a ) > , (3.18) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) (cid:26)(cid:16)(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12)(cid:17) α ( α + 1) ( α + 2) (cid:27) . Theorem 5.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Let f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is P ϕ − preinvex on I , then, for η ( b, a ) > , (3.19) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) (cid:18) − α (cid:19) h(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q i q Z h ( t ) dt q Proof.
Using Lemma 3, we have (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) e iϕ η ( b, a ) (cid:3) Z " − (1 − t ) α +1 − t α +1 α + 1 f ′′ ( a + te iϕ η ( b, a )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) Z " − (1 − t ) α +1 − t α +1 α + 1 p dt p Z (cid:12)(cid:12)(cid:12) f ′′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) q dt q ≤ (cid:2) e iϕ η ( b, a ) (cid:3) α + 1 (cid:18) − α (cid:19) Z n h (1 − t ) (cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + h ( t ) (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q o dt q = (cid:2) e iϕ η ( b, a ) (cid:3) α + 1 (cid:18) − α (cid:19) n(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q o Z [ h ( t )] dt q This completes the proof. (cid:3)
We have some special cases for (3 . I. If h ( t ) = t , then we have the result for ϕ − preinvexity. Corollary 13.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Suppose that f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is ϕ − preinvex on I, then, for η ( b, a ) > , (3.20) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) α + 1 (cid:18) − α (cid:19) (cid:18) (cid:19) q (cid:16)(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q (cid:17) q II. If h ( t ) = t s , then we have the result for s ϕ − preinvexity Corollary 14.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Let f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is s ϕ − preinvex on I , then, for η ( b, a ) > and s ∈ (0 , , (3.21) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) α + 1 (cid:18) − α (cid:19) (cid:18) s + 1 (cid:19) q (cid:16)(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q (cid:17) q III. If h ( t ) = 1, then we have the result for P ϕ − preinvexity. Corollary 15.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Let f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is P ϕ − preinvex on I , then, for η ( b, a ) > , (3.22) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) α + 1 (cid:18) − α (cid:19) (cid:16)(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q (cid:17) q Theorem 6.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Let f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is P ϕ − preinvex on I , then, for η ( b, a ) > , (3.23) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) (cid:18) α ( α + 1) ( α + 2) (cid:19) p × n(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q o Z " − (1 − t ) α +1 − t α +1 α + 1 h ( t ) dt q ERMITE–HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE ... Proof.
Using Lemma 3 and well-known power-mean inequality, we have (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) e iϕ η ( b, a ) (cid:3) Z " − (1 − t ) α +1 − t α +1 α + 1 f ′′ ( a + te iϕ η ( b, a )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) Z " − (1 − t ) α +1 − t α +1 α + 1 dt p × Z " − (1 − t ) α +1 − t α +1 α + 1 f ′′ ( a + te iϕ η ( b, a )) (cid:12)(cid:12)(cid:12) q dt q ≤ (cid:2) e iϕ η ( b, a ) (cid:3) Z " − (1 − t ) α +1 − t α +1 α + 1 dt p × Z " − (1 − t ) α +1 − t α +1 α + 1 h (1 − t ) (cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + h ( t ) (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q o dt q ≤ (cid:2) e iϕ η ( b, a ) (cid:3) (cid:18) α ( α + 1) ( α + 2) (cid:19) p Z " − (1 − t ) α +1 − t α +1 α + 1 × (cid:16) h (1 − t ) (cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + h ( t ) (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q (cid:17) dt o q = (cid:2) e iϕ η ( b, a ) (cid:3) (cid:18) α ( α + 1) ( α + 2) (cid:19) p nn(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q o × Z " − (1 − t ) α +1 − t α +1 α + 1 h ( t ) dt q This completes the proof. (cid:3)
We have some special cases for (3 . I. If h ( t ) = t , then we have the result for ϕ − preinvexity. Corollary 16.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Suppose that f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is ϕ − preinvex on I, then, for every η ( b, a ) > , (3.24) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) (cid:18) (cid:19) q (cid:18) α ( α + 1) ( α + 2) (cid:19) (cid:16)n(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q o(cid:17) q II. If h ( t ) = t s , then we have the result for s ϕ − preinvexity Corollary 17.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Let f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is s ϕ − preinvex on I , then, for every η ( b, a ) > and s ∈ (0 , , (3.25) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) (cid:18) α ( α + 1) ( α + 2) (cid:19) p × n(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q o α + s + 2) ( s + 1) − B ( s + 1 , α ) α + 1 !! q III. If h ( t ) = 1, then we have the result for P ϕ − preinvexity. Corollary 18.
Let I ⊆ R be a open invex set with respect to η : I × I → R . Let f : I → R be a twice differentiable function such that f ′′ ∈ L [ a, a + e iϕ η ( b, a )] . If (cid:12)(cid:12)(cid:12) f ′′ (cid:12)(cid:12)(cid:12) is P ϕ − preinvex on I , then, for every η ( b, a ) > , (3.26) (cid:12)(cid:12)(cid:12)(cid:12) f ( a ) + f ( a + e iϕ η ( b, a )) − Γ( α + 1)[ e iϕ η ( b, a )] α n J αa + f ( x ) + J α ( a + e iϕ η ( b,a )) − f ( x ) o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:2) e iϕ η ( b, a ) (cid:3) (cid:18) α ( α + 1) ( α + 2) (cid:19) (cid:16)(cid:12)(cid:12)(cid:12) f ′′ ( a ) (cid:12)(cid:12)(cid:12) q + (cid:12)(cid:12)(cid:12) f ′′ ( b ) (cid:12)(cid:12)(cid:12) q (cid:17) q Conclusion 1.
If we take α = 1 in (3 . − (3 . , (3 . − (3 . , (3 . − (3 . , (3 . − (3 . , (3 . − (3 . , (3 . − (3 . , we obtain Crollary . − . in [20] . Conclusion 2.
If we take ϕ = 0 in (3 . , we obtain Lemma . in [4] . Conclusion 3.
If we take α = 1 and ϕ = 0 in our results, we get some Hermite-Hadamard inequalities. ERMITE–HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE ... References [1] Barani, A.; Ghazanfari, A.G.; Dragomir, S.S.: Hermite-Hadamard inequality for functionswhose derivatives absolute values are preinvex. J. Inequal. Appl. 2012, 247 (2012)[2] Dragomir, S.S.; Pearce, C.E.M.: Selected topics on Hermite–Hadamard inequalities and ap-plications. Victoria University (2000)[3] Dragomir, S.S.; Pecaric, J.; Persson, L.E.: Some inequalities of Hadamard type. Soochow J.Math. 21, 335–341, (1995)[4] ˙I¸scan, ˙I.: Hermite–Hadamard’s inequalities for preinvex function via fractional integrals andrelated functional inequalities, Am. J. Math. Anal. 1 (3) (2013) 33–38.[5] Latif, M.A.: Some inequalities for differentiable prequasiinvex functions with applications.Konuralp J. Math. 1(2), 17–29 (2013)[6] Latif, M.A.; Dragomir, S.S.: Some Hermite-Hadamard type inequalities for functions whosepartial derivatives in absloute value are preinvex on the co-oordinates. Facta Universitatis(NIS) Ser. Math. Inform. 28(3), 257–270, (2013)[7] Latif, M.A.; Dragomir, S. S.; Momoniat, E.: Some weighted integral inequalities for differen-tiable preinvex and prequasiinvex functions. RGMIA (2014)[8] Noor, M.A.: On Hadamard integral inequalities involving two log-preinvex functions. J. In-equal. Pure Appl. Math. 8, 1–6 (2007)[9] Noor, M.A.: On Hermite-Hadamard integral inequalities for product of two nonconvex func-tions. J. Adv. Math. Studies 2(1), 53–62 (2009)[10] Noor, M.A.: Some new classes of nonconvex functions. Nonl. Funct. Anal. Appl. 11(1), 165–171 (2006)[11] Noor, M.A.; Awan, M.U.; Noor, K.I.: On some inequalities for relative semi-convex functions.J. Inequal. Appl., 2013, 332 (2013)[12] Noor, M.A.; Noor, K.I.: Generalized preinvex functions and their properties. Journal of Appl.Math. Stochastic Anal., 2006(12736), 1–13, doi:10.1155/JAMSA/2006/12736[13] Noor, M.A.; Noor, K.I.; Ashraf, M.A.; Awan, M.U.; Bashir, B.: Hermite–Hadamard inequal-ities for h ϕ − convex functions. Nonl. Anal. Formum. 18, 65–76 (2013)[14] Noor, M.A.; Noor, K.I.; Awan, M.U.: Hermite–Hadamard inequalities for relative semi-convex functions and applications. Filomat, 28(2), 221–230, (2014)[15] Noor, M.A.; Noor, K.I.; Awan, M.U.: Generalized convexity and integral inequalities. Appl.Math. Inf. Sci. 9(1), 233–243, (2015)[16] Noor, M.A.; Noor, K.I.; Awan, M.U.; Khan, S.: Fractional Hermite-Hadamard inequalitiesfor some new classes of Godunova–Levin functions. Appl. Math. Inf. Sci. 8(6), 2865–2872,(2014)[17] Noor, M.A.; Noor, K.I.; Awan, M.U.; Li, J.: On Hermite–Hadamard type Inequalities forh-preinvex functions. Filomat, in press[18] Noor, M.A.; Noor, K.I.; Al-Said, E.: Iterative methods for solving nonconvex equilibriumvariational inequalities. Appl. Math. Inf. Sci. 6(1), 65–69, (2012)[19] Noor, M.A.; Postolache, M.; Noor, K.I.; Awan, M.U.: Geometrically nonconvex functionsand integral inequalities. Appl. Math. Inf. Sci. 9, (2015)[20] Noor, M.A.; Noor, K.I.; Awan, M.A.; Khan, S.: Hermite–Hadamard type inequalities fordifferantiable h ϕ − preinvex functions. Arab. J. Math.4:63-76 (2015)[21] Sarikaya, M.Z.; Set, E.: ¨Ozdemir M.E.: On some new inequalities of Hadamard type involvingh-convex functions. Acta Math. Univ. Comenianae. 2, 265–272 (2010)[22] Sarikaya, M.Z.; Alp, N.; Bozkurt, H.: On Hermite-Hadamard Type Integral Inequalitiesfor preinvex and log-preinvex functions, Contemporary Analysis and Applied Mathematics,Vol.1, No.2, 237-252, 2013.[23] Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326, 303–311 (2007)[24] Weir, T.; Mond, B.: Preinvex functions in multiple objective optimization. J. Math. Anal.Appl. 136, 29–38 (1998)[25] Samko, S.G.; Kilbas, A.A.; Marichev, O.I.: Fractional Integrals and Derivatives, Theory andApplications, Gordon and Breach, Yverdon, Switzerland, 1993 [Department of Mathematics, Faculty of Science and Arts, University of Kahramanmaras¸S¨utc¸¨u ˙Imam, 46000, Kahramanmaras¸, Turkey E-mail address : [email protected] [Department of Mathematics, Faculty of Science and Arts, University of Kahramanmaras¸S¨utc¸¨u ˙Imam, 46000, Kahramanmaras¸, Turkey E-mail address ::