Hermite--Hadamard type inequalities via differentiable $ h_\varphi ...

arXiv:1512.03730v1 [math.CA] 29 Nov 2015

HERMITE–HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hϕ −PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS ¨ ABDULLAH AKKURT AND HUSEYIN YILDIRIM Abstract. In this paper, we consider a new class of convex functions which is called hϕ −preinvex functions. We prove several Hermite–Hadamard type inequalities for differentiable hϕ −preinvex functions via Fractional Integrals. Some special cases are also discussed. Our results extend and improve the corresponding ones in the literature.

1. Introduction The following inequality is well-known in the literature as Hermite-Hadamard inequality. Let f : I ⊆ R → R be a convex function with a < b and a, b ∈ I. Then the following holds (1.1)

1 a+b )≤ f( 2 b−a

Zb

f (x)dx ≤

f (a) + f (b) . 2

a

Recently, Hermite-Hadamard type inequality has been the subject of intensive research. Hermite–Hadamard inequality can be considered as necessary and sufficient condition for a function to be convex. It provides estimates of the mean value of continuous convex function. In recent years, several extensions and generalizations have been proposed for classical convexity (see [1 − 24]). A significant generalization of classical convex functions is that of ϕ−convex functions introduced by Noor [10]. Noor [10] has investigated the basic properties of ϕ−convex functions and showed that ϕ−convex functions are nonconvex functions. Noor [9] extended Hermite–Hadamard type inequalities for ϕ−convex functions. Motivated by the ongoing research on generalizations and extensions of classical convexity, Varosanec [23] introduced the class of h−convex functions. Noor et al. [13] introduced another generalization of classical convexity which is called as hϕ −convex functions. Finally Noor et al. Several Hermite–Hadamard type inequalities are proved for hϕ −preinvex functions[20]. Weir and Mond [24] investigated the class of preinvex functions. It is well-known that the preinvex functions and invex sets may not be convex functions and convex sets. For recent investigation on preinvexity (see [1, 4, 5 − 8, 17, 22]). Motivated and inspired by the recent research in this field, we consider a new class of convex functions, which is called hϕ −preinvex functions. This class includes Key words and phrases. integral inequalities, fractional integrals, Hermite-Hadamard Inequality, Preinvex functions, H¨ older’s inequality. 2010 Mathematics Subject Classification 26D15, 26A51, 26A33, 26A42. 1

2

¨ ABDULLAH AKKURT AND HUSEYIN YILDIRIM

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 several new Hermite–Hadamard type fractional integral inequalities for hϕ −preinvex functions and their variant forms. Results obtained in this paper continue to hold for these special cases. Our results represent significant generalized of the previous results. The interested readers are encouraged to find novel and innovative applications of hϕ −preinvex functions and fractional integrals. 2. Preliminares π be a Let Rn be the finite dimensional Euclidian space. Also let 0 ≤ ϕ ≤ 2 continuous function. Definition 1. ([12]) The set Kϕn in Rn is said to be ϕ−invex at u with respect to ϕ(.), if there exists a bifunction η(., .) : Kϕn × Kϕn → R, such that (2.1)

u + teiϕ η(v, u) ∈ Kϕn ,

∀u, v ∈ Kϕn , t ∈ [0, 1].

The ϕ−invex set Kϕn 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 Kϕ be a set in Rn . Then the set Kϕ is said to be ϕ−convex with respect to ϕ, if and only if u + teiϕ (v, u) ∈ Kϕ ,

∀u, v ∈ Kϕ , t ∈ [0, 1].

For ϕ = 0, the set Kϕ reduces to the classical convex set K. That is, u + t(v, u) ∈ K,

∀u, v ∈ K, t ∈ [0, 1].

Definition 3. ([24]) A set Kn is said to be invex set with respect to bifunction η(., .), if u + tη(v, u) ∈ Kn , ∀u, v ∈ Kn , t ∈ [0, 1]. The invex set Kn is also called η−connected set. Definition 4. ([20])Let h : J ⊆ R → R be a nonnegative function. A function f on the set Kϕn is said to be hϕ −preinvex function with respect to ϕ and bifunction η, if (2.2)

f (u + teiϕ η(v, u)) ≤ h(1 − t)f (u) + h(t)f (v),

∀u, v ∈ Kϕn , t ∈ [0, 1].

Remark 1. One can deduce several known concepts from Definition 4 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 hϕ −convex functions (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) = ts where s ∈ (0, 1] in (2.2) we have the definition for sϕ −preinvex functions.

HERMITE–HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE...

3

Definition 5. A function f on the set Kϕn is said to be sϕ −preinvex function with respect to ϕ and η, if (2.3)

f (u + teiϕ η(v, u)) ≤ (1 − t)s f (u) + ts f (v),

∀u, v ∈ Kϕn , t ∈ [0, 1].

II. For h(t) = 1 in (2.2) we have the definition for Pϕ −preinvex functions. Definition 6. A function f on the set Kϕn is said to be sϕ −preinvex function with respect to ϕ and η, if (2.4)

f (u + teiϕ η(v, u)) ≤ f (u) + f (v),

∀u, v ∈ Kϕn , t ∈ [0, 1].

Definition 7. ([25]) Let f ∈ L1 [a, b]. The Riemann-Liouville fractional integral Jaα+ f (x) and Jbα− f (x) of order α > 0 are defined by

Jaα+

(2.5)

[f (x)] =

1 Γ(α)

Zx

(x − t)

f (t) dt

x>a

Zb

(t − x)α−1 f (t) dt

x 0. If f ∈ L1 [a, a + eiϕ η(b, a)] and α ≥ 0, then (3.1) o Γ(α + 1) n α α f (a) + f (a + eiϕ η(b, a)) − iϕ α Ja+ f (x) + J(a+eiϕ η(b,a))− f (x) [e η(b, a)] Z1 ′ = eiϕ η(b, a) [tα − (1 − t)α ] f (a + teiϕ η(b, a))dt. 0


4

Proof. Let Z1

α

′

[(1 − t) − tα ] f (a + teiϕ η(b, a))dt =

0

1 α ((1 − t) − tα ) f (a + teiϕ η(b, a)) eiϕ η(b, a) 0

Z1 h i α (1 − t)α−1 + tα−1 f (a + teiϕ η(b, a))dt iϕ e η(b, a) 0 Z1 h i f (a) + f (a + eiϕ η(b, a)) α α−1 α−1 =− f (a + teiϕ η(b, a))dt (1 − t) + t + eiϕ η(b, a) eiϕ η(b, a) 0 1 = − eiϕ η(b,a) f (a) + f (a + eiϕ η(b, a))  a+eiϕ Zη(b,a) α−1 α−1  du + eiϕu−a f (u) eiϕ η(b,a) −α 1 − eiϕu−a  η(b,a) η(b,a) +

a f (a) + f (a + eiϕ η(b, a)) =− eiϕ η(b, a) a+eiϕ Zη(b,a) α−1 u−a α−1 α iϕ a + e η(b, a) − u + eiϕ η(b,a) f (u) du + iϕ e η(b, a) a i f (a) + f (a + eiϕ η(b, a)) Γ (α + 1) h α α J f (x) + J f (x) . =− + + − iϕ a (a+e η(b,a)) eiϕ η(b, a) [eiϕ η(b, a)]α+1

This complete the proof.

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 ∈ L1 [a, a + ′ eiϕ η(b, a)]. Let f be hϕ −preinvex function. Then, for η(b, a) > 0 and α > 0, (3.2)

h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] h ′ ′ i Z1 α iϕ ≤ e η(b, a) f (a) + f (b) |(1 − t) − tα | h(t)dt. 0

′ Proof. Using Lemma 2 and hϕ −preinvexity of f , we have

i Γ(α + 1) h α α J f (x) + J f (x) α a+ (a+eiϕ η(b,a))− [eiϕ η(b, a)] Z1 ′ ≤ eiϕ η(b, a) |(1 − t)α − tα | f (a + teiϕ η(b, a)) dt

f (a) + f (a + eiϕ η(b, a)) − 0

≤ eiϕ η(b, a)

Z1 0

′ ′ α |(1 − t) − tα | h (1 − t) f (a) + h (t) f (b) dt


5

1 Z ′ iϕ  = e η(b, a) |(1 − t)α − tα | h (1 − t) f (a) dt 0

+

Z1

 ′ α |(1 − t) − tα | h (t) f (a) dt

Z1

 ′ α |(1 − t) − tα | h (t) f (b) dt

1 0 Z ′ = eiϕ η(b, a)  |tα − (1 − t)α | h (t) f (a) dt 0

+

0

h ′ ′ i Z1 α iϕ = e η(b, a) f (a) + f (b) |(1 − t) − tα | h(t)dt. 0

This completes the proof.

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. Suppose ′ iϕ that ′ f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is ϕ−preinvex on I, then, for η(b, a) > 0, h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) − a (a+e η(b,a)) [eiϕ η(b, a)] (3.3) h ′ ′ i 1 1 ≤ eiϕ η(b, a) f (a) + f (b) 1 − α+2 . α+2 2 II. If h(t) = ts , then we have the result for sϕ −preinvexity

Corollary 2. Let I ⊆ R be a open invex set with respect to η : I × I → R. Suppose ′ iϕ that ′ f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is sϕ −preinvex on I, then, for η(b, a) > 0 and s ∈ (0, 1], h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a iϕ (a+e η(b,a))− h ′[e η(b, a)] i h ′ ≤ eiϕ η(b, a) f (a) + f (b) B 21 (α + 1, s + 1) (3.4) 1 1 −B 21 (s + 1, α + 1) + 1 − s+α . α+s+1 2 III. If h(t) = 1, then we have the result for Pϕ −preinvexity. Corollary 3. Let I ⊆ R be a open invex set with respect to η : I × I → R. Suppose ′ iϕ that ′ f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is Pϕ −preinvex on I, then, for η(b, a) > 0, h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] (3.5) h ′ ′ i 2 1 ≤ eiϕ η(b, a) f (a) + f (b) 1− α . α+1 2


6

Theorem 2. Let I ⊆ R be a open invex set with respect to η : I × I → R. Suppose ′ iϕ that ′ p f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. Let f be hϕ −preinvex on I, p1 + q1 = 1 and η(b, a) > 0. Then (3.6) h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)]  p−1  1 p1 p1  p Z p  p 1 2 ′ p−1 ′ p−1 iϕ h(t)dt ≤ e η(b, a) 1 − αp + f (b) . f (a)   αp + 1 2 0

Proof. Using Lemma 2, we have h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− iϕ η(b, a)] [e Z1 iϕ ′ α α iϕ = e η(b, a) [(1 − t) − t ] f (a + te η(b, a))dt 0 Z1 ′ ≤ eiϕ η(b, a) |(1 − t)α − tα | f (a + teiϕ η(b, a)) dt. 0

From Holders inequality, we have h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)]  1q 1  p1  1 Z Z q ′ p ≤ eiϕ η(b, a)  |(1 − t)α − tα | dt  f (a + teiϕ η(b, a)) dt 0

0



 = eiϕ η(b, a) 

1 2

Z

α

p

((1 − t) − tα ) dt +

0

Z1 1 2

 p1

 α p (tα − (1 − t) ) dt

1  q1 Z q ′ ×  f (a + teiϕ η(b, a)) dt . 0

′ p Here hϕ −preinvexity of f , we have following inequality h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)]   q1 1 Z1 h 1 i 2p 1 p ′ q ′ q iϕ ≤ η(b, a) 1 − αp h(1 − t) f (a) + h(t) f (b) dt 1 e 2 (αp + 1) p 0  p−1  1 p1 1  p Z p  p ′ p−1 p−1 2 1 ′ p iϕ h(t)dt = + f (b) e η(b, a) 1 − αp . f (a)   αp + 1 2 0


Now we have some special cases for (3.6).


7

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. Suppose ′ iϕ that ′ p f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. Let 1 1 f be ϕ−preinvex on I, p + q = 1 and η(b, a) > 0. Then (3.7)

h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)]  p p  p−1 p ′ p−1 ′ p−1  p1 1    (a) + (b) f f 1 p 2 . eiϕ η(b, a) 1 − αp ≤   αp + 1 2 2  

II. If h(t) = ts , then we have the result for sϕ −preinvexity

Corollary 5. Let I ⊆ R be a open invex set with respect to η : I × I → R. Suppose ′ iϕ that ′ p f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. Let f be sϕ −preinvex on I, p1 + 1q = 1, η(b, a) > 0 and s ∈ (0, 1]. Then (3.8)

h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] p−1  p p  p ′ p−1 ′ p−1  p1 1   f (a)  + f (b) 2 1 p ≤ . eiϕ η(b, a) 1 − αp   αp + 1 2 s+1  

III. If h(t) = 1, then we have the result for Pϕ −preinvexity.

Corollary 6. Let I ⊆ R be a open invex set with respect to η : I × I → R. Suppose ′ iϕ that ′ p f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. Let 1 1 f be Pϕ −preinvex on I, p + q = 1 and η(b, a) > 0. Then (3.9)

h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] p−1 p1 1 p p 2 1 p ′ p−1 ′ p−1 p iϕ (a) ≤ e η(b, a) 1 − αp + . (b) f f αp + 1 2

Theorem 3. Let I ⊆ R be a open invex set with respect to η : I × I → R. Suppose ′ iϕ that ′ q f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. Let f , q > 1 is hϕ −preinvex on I, then, for η(b, a) > 0,

(3.10) h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)]  p−1  1 p1 1 p Z p  p  ′ p−1 ′ p−1 2 1 p iϕ α α ≤ |(1 − t) − t | h(t)dt 1− α e η(b, a) + f (b) . f (a)   α+1 2 0


8

Proof. Using Lemma 2, we have h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] Z1 iϕ ′ α = e η(b, a) [(1 − t) − tα ] f (a + teiϕ η(b, a))dt 0 Z1 ′ α ≤ eiϕ η(b, a) |(1 − t) − tα | f (a + teiϕ η(b, a)) dt 0

Using power-mean inequality, we have h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)]  q1 1  p1  1 Z Z q ′ ≤ eiϕ η(b, a)  |(1 − t)α − tα | dt  f (a + teiϕ η(b, a)) dt 0

0



 = eiϕ η(b, a) 

1 2

Z

α

((1 − t) − tα ) dt +

0

Z1 1 2

 p1

 α (tα − (1 − t) ) dt

1  1q Z q ′ ×  f (a + teiϕ η(b, a)) dt 0

′ q now using hϕ −preinvexity of f , we have h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] 1  q1 1 1 Z h ′ q ′ q i 2p 1 p iϕ ≤ h(1 − t) f (a) + h(t) f (b) dt η(b, a) 1 − α 1 e 2 (α + 1) p 0  p−1  1 p1 p1  p Z p  p ′ p−1 p−1 1 2 ′ α α iϕ |(1 − t) − t | h(t)dt = e η(b, a) 1 − α + f (b) f (a)   α+1 2 0


Now we have some special cases for (3.10). 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. Suppose ′ iϕ that ′ q f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f , q > 1 is ϕ−preinvex on I, then, for η(b, a) > 0, (3.11) h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] p1 1 p−1 p 1 p p 1 2 1 p iϕ ′ p−1 ′ p−1 1 − α+2 ≤ 1− α e η(b, a) f (a) + f (b) . α+1 2 α+2 2


9

II. If h(t) = ts , then we have the result for sϕ −preinvexity Corollary 8. Let I ⊆ R be a open invex set with respect to η : I × I → R. Suppose ′ iϕ that ′ q f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f , q > 1 is sϕ −preinvex on I, then, for η(b, a) > 0 and s ∈ (0, 1], (3.12) h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] α+1 p p ′ p−1 ′ p−1 2 −2 iϕ e η(b, a) (a) ≤ + (b) f f 2α (α + 1) p−1 p 1 1 . 1 − s+α x B 21 (α + 1, s + 1) − B 21 (s + 1, α + 1) + α+s+1 2 III. If h(t) = 1, then we have the result for Pϕ −preinvexity. Corollary 9. Let I ⊆ R be a open invex set with respect to η : I × I → R. Suppose ′ iϕ that ′ q f : I → R is a differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f , q > 1 is Pϕ −preinvex on I, then, for η(b, a) > 0 (3.13) h i f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] α+1 p p p−1 2 −2 ′ p−1 ′ p−1 p iϕ e η(b, a) f (a) ≤ + f (b) . 2α (α + 1)

Using the technique of [1], we prove the following result which helps us in proving our next results. Lemma 2. Let I ⊆ R be a open invex set with respect to η : I × I → R where ′′ η(b, a) > 0. If f ∈ L1 [a, a + eiϕ η(b, a)], then o Γ(α + 1) n α α J f (x) + J f (x) f (a) + f (a + eiϕ η(b, a)) − iϕ α a+ (a+eiϕ η(b,a))− [e η(b, a)] # 1 " Z (3.14) 2 1 − (1 − t)α+1 − tα+1 ′′ f (a + teiϕ η(b, a))dt. = eiϕ η(b, a) α+1 0

Proof. Let # Z1 " α+1 1 − (1 − t) − tα+1 ′′ f (a + teiϕ η(b, a))dt α+1 0 1 Z1 ′ 1 1 − (1 − t)α+1 − tα+1 f (a + teiϕ η(b, a)) ′ α [(1 − t) − tα ] f (a + teiϕ η(b, a))dt = − iϕ iϕ α+1 e η(b, a) e η(b, a) 0

=−

=

1

eiϕ η(b, a)

Z1

1 2

(eiϕ η(b, a))

0

α

0

′

[(1 − t) − tα ] f (a + teiϕ η(b, a))dt

f (a) + f (a + e η(b, a)) −


iϕ

o Γ(α + 1) n α α α Ja+ f (x) + J(a+eiϕ η(b,a))− f (x) [eiϕ η(b, a)]

10


Theorem 4. Let I ⊆ R be a open invex set with respect to η : I × I → R. Let ′′ iϕ f :′′ I → R be a twice differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f , is hϕ −preinvex on I, then, for η(b, a) > 0, (3.15) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a iϕ η(b, a)] (a+e η(b,a))− [e   # Z1 "  iϕ 2  ′′ ′′ 1 − (1 − t)α+1 − tα+1 ≤ e η(b, a) h(t)dt . f (a) + f (b)   α+1 0

Proof. Using Lemma 3, we have n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− iϕ η(b, a)] [e # Z1 " α+1 α+1 iϕ 2 1 − (1 − t) − t ′′ iϕ = e η(b, a) f (a + te η(b, a))dt α + 1 0 # Z1 " 2 1 − (1 − t)α+1 − tα+1 ′′ ≤ eiϕ η(b, a) f (a + teiϕ η(b, a)) dt α+1 0 " # 1 Z α+1 ′′ ′′ iϕ 2 1 − (1 − t) − tα+1 ≤ e η(b, a) h(1 − t) f (a) + h(t) f (b) dt α+1 0 # Z1 " iϕ 2 ′′ ′′ 1 − (1 − t)α+1 − tα+1 = e η(b, a) h(t)dt f (a) + f (b) α+1 0


Now, we discuss some special cases for (3.15). 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 ∈ ′′ L1 [a, a + eiϕ η(b, a)]. If f is ϕ−preinvex on I, then, for η(b, a) > 0, (3.16) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] ′′ ′′ 2 α ≤ eiϕ η(b, a) . f (a) + f (b) 2 (α + 1) (α + 2) II. If h(t) = ts , then we have the result for sϕ −preinvexity

Corollary 11. Let I ⊆ R be a open invex set with respect to η : I × I → R. Let ′′ iϕ f :′′ I → R be a twice differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is sϕ −preinvex on I, then, for η(b, a) > 0 and s ∈ (0, 1], (3.17) n o Γ(α + 1) α α iϕ f (a) + f (a + e η(b, a)) − Ja+ f (x) + J(a+eiϕ η(b,a))− f (x) iϕ η(b, a)]α [e ( !) B 12 (s + 1, α) iϕ 2 ′′ ′′ 1 ≤ e η(b, a) . − f (a) + f (b) (α + s + 2) (s + 1) α+1


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III. If h(t) = 1, then we have the result for Pϕ −preinvexity. Corollary 12. Let I ⊆ R be a open invex set with respect to η : I × I → R. Let ′′ iϕ f :′′ I → R be a twice differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is Pϕ −preinvex on I, then, for η(b, a) > 0, (3.18) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a iϕ η(b, a)] (a+e η(b,a))− [e 2 ′′ ′′ α . ≤ eiϕ η(b, a) f (a) + f (b) (α + 1) (α + 2)

Theorem 5. Let I ⊆ R be a open invex set with respect to η : I × I → R. Let ′′ iϕ f :′′ I → R be a twice differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is Pϕ −preinvex on I, then, for η(b, a) > 0,

(3.19) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] 1  q1 Z 1 iϕ 2 1 h ′′ q ′′ q i q  1− α ≤ e η(b, a) h(t)dt f (a) + f (b) 2 0

Proof. Using Lemma 3, we have

n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a iϕ η(b, a)] (a+e η(b,a))− [e # Z1 " α+1 α+1 iϕ 2 1 − (1 − t) − t ′′ iϕ = e η(b, a) f (a + te η(b, a))dt α + 1 0 1 "  1q  p1  1 # p Z Z α+1 α+1 q 2 1 − (1 − t) −t ′′ ≤ eiϕ η(b, a)  dt  f (a + teiϕ η(b, a)) dt α+1 0 0  q1  iϕ 2 Z1 n o e η(b, a) 1 ′′ q ′′ q ≤ 1− α  h(1 − t) f (a) + h(t) f (b) dt α+1 2 0  q1  1 iϕ 2 n Z o e η(b, a) 1 ′′ q ′′ q = 1 − α  f (a) + f (b) [h(t)] dt α+1 2 0


We have some special cases for (3.19). 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 ∈

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′′ L1 [a, a + eiϕ η(b, a)]. If f is ϕ−preinvex on I, then, for η(b, a) > 0, (3.20) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] iϕ 2 q1 1 e η(b, a) 1 1 ′′ q ′′ q q (a) + (b) ≤ 1− α f f α+1 2 2 II. If h(t) = ts , then we have the result for sϕ −preinvexity

Corollary 14. Let I ⊆ R be a open invex set with respect to η : I × I → R. Let ′′ iϕ f :′′ I → R be a twice differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is sϕ −preinvex on I, then, for η(b, a) > 0 and s ∈ (0, 1], (3.21) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] iϕ 2 q1 1 e η(b, a) 1 1 ′′ q ′′ q q ≤ 1− α f (a) + f (b) α+1 2 s+1 III. If h(t) = 1, then we have the result for Pϕ −preinvexity.

Corollary 15. Let I ⊆ R be a open invex set with respect to η : I × I → R. Let ′′ iϕ f :′′ I → R be a twice differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is Pϕ −preinvex on I, then, for η(b, a) > 0, (3.22) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] iϕ 2 1 e η(b, a) 1 ′′ q ′′ q q 1− α ≤ f (a) + f (b) α+1 2 Theorem 6. Let I ⊆ R be a open invex set with respect to η : I × I → R. Let ′′ iϕ f :′′ I → R be a twice differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is Pϕ −preinvex on I, then, for η(b, a) > 0, (3.23) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] p1 iϕ 2 α ≤ e η(b, a) (α + 1) (α + 2)  1q  # " n ′′ q ′′ q o Z1 1 − (1 − t)α+1 − tα+1 h(t)dt ×  f (a) + f (b) α+1 0


Proof. Using Lemma 3 and well-known power-mean inequality, we have n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] # Z1 " α+1 2 1 − (1 − t) − tα+1 ′′ = eiϕ η(b, a) f (a + teiϕ η(b, a))dt α+1 0 1 1 "  # p Z α+1 iϕ 2 1 − (1 − t) − tα+1   ≤ e η(b, a) dt α+1 0 1 "  q1 # Z α+1 q α+1 1 − (1 − t) − t ′′ × f (a + teiϕ η(b, a)) dt α+1 0 1 " #  p1 Z α+1 α+1 iϕ 2 1 − (1 − t) − t ≤ e η(b, a)  dt α+1 0 1 "  q1 # Z α+1 ′′ q ′′ q o α+1 n 1 − (1 − t) − t × h(1 − t) f (a) + h(t) f (b) dt α+1 0  # p1 Z1 " iϕ 2 α 1 − (1 − t)α+1 − tα+1 ≤ e η(b, a)  (α + 1) (α + 2) α+1 0 ′′ q ′′ q o q1 × h(1 − t) f (a) + h(t) f (b) dt p1 nn o iϕ 2 α ′′ q ′′ q = e η(b, a) f (a) + f (b) (α + 1) (α + 2)  q1 # Z1 " α+1  α+1 1 − (1 − t) −t h(t)dt ×  α+1

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0


We have some special cases for (3.23). 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 ∈ ′′ L1 [a, a + eiϕ η(b, a)]. If f is ϕ−preinvex on I, then, for every η(b, a) > 0, (3.24) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] 1 n o 1 iϕ 2 1 q α ′′ q ′′ q q ≤ e η(b, a) f (a) + f (b) 2 (α + 1) (α + 2) II. If h(t) = ts , then we have the result for sϕ −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 ∈ L1 [a, a + eiϕ η(b, a)]. If

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′′ f is sϕ −preinvex on I, then, for every η(b, a) > 0 and s ∈ (0, 1], (3.25) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] p1 2 α ≤ eiϕ η(b, a) (α + 1) (α + 2) !! q1 n ′′ q ′′ q o B 21 (s + 1, α) 1 − × f (a) + f (b) (α + s + 2) (s + 1) α+1 III. If h(t) = 1, then we have the result for Pϕ −preinvexity.

Corollary 18. Let I ⊆ R be a open invex set with respect to η : I × I → R. Let ′′ iϕ f :′′ I → R be a twice differentiable function such that f ∈ L1 [a, a + e η(b, a)]. If f is Pϕ −preinvex on I, then, for every η(b, a) > 0, (3.26) n o f (a) + f (a + eiϕ η(b, a)) − Γ(α + 1) α J α+ f (x) + J α iϕ f (x) a (a+e η(b,a))− [eiϕ η(b, a)] 1 q q iϕ 2 α ′′ ′′ q ≤ e η(b, a) f (a) + f (b) (α + 1) (α + 2) Conclusion 1. If we take α = 1 in (3.3)−(3.5), (3.7)−(3.9), (3.11)−(3.13), (3.16)− (3.18), (3.20) − (3.22), (3.24) − (3.26), we obtain Crollary 3.3 − 3.26 in [20]. Conclusion 2. If we take ϕ = 0 in (3.1), we obtain Lemma 2.3 in [4]. Conclusion 3. If we take α = 1 and ϕ = 0 in our results, we get some HermiteHadamard inequalities.


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References [1] Barani, A.; Ghazanfari, A.G.; Dragomir, S.S.: Hermite-Hadamard inequality for functions whose 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 applications. Victoria University (2000) [3] Dragomir, S.S.; Pecaric, J.; Persson, L.E.: Some inequalities of Hadamard type. Soochow J. Math. 21, 335–341, (1995) ˙scan, I.: ˙ Hermite–Hadamard’s inequalities for preinvex function via fractional integrals and [4] I¸ related 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 whose partial 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 differentiable preinvex and prequasiinvex functions. RGMIA (2014) [8] Noor, M.A.: On Hadamard integral inequalities involving two log-preinvex functions. J. Inequal. Pure Appl. Math. 8, 1–6 (2007) [9] Noor, M.A.: On Hermite-Hadamard integral inequalities for product of two nonconvex functions. 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 inequalities for hϕ −convex functions. Nonl. Anal. Formum. 18, 65–76 (2013) [14] Noor, M.A.; Noor, K.I.; Awan, M.U.: Hermite–Hadamard inequalities for relative semiconvex 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 inequalities for 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 for h-preinvex functions. Filomat, in press [18] Noor, M.A.; Noor, K.I.; Al-Said, E.: Iterative methods for solving nonconvex equilibrium variational inequalities. Appl. Math. Inf. Sci. 6(1), 65–69, (2012) [19] Noor, M.A.; Postolache, M.; Noor, K.I.; Awan, M.U.: Geometrically nonconvex functions and integral inequalities. Appl. Math. Inf. Sci. 9, (2015) [20] Noor, M.A.; Noor, K.I.; Awan, M.A.; Khan, S.: Hermite–Hadamard type inequalities for differantiable hϕ −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 involving h-convex functions. Acta Math. Univ. Comenianae. 2, 265–272 (2010) [22] Sarikaya, M.Z.; Alp, N.; Bozkurt, H.: On Hermite-Hadamard Type Integral Inequalities for 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 and Applications, Gordon and Breach, Yverdon, Switzerland, 1993

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[Department of Mathematics, Faculty of Science and Arts, University of Kahramanmaras¸ ˙ ¨ tc ¨ Imam, Su ¸u 46000, Kahramanmaras¸, Turkey E-mail address: [email protected] [Department of Mathematics, Faculty of Science and Arts, University of Kahramanmaras¸ ˙ ¨ tc ¨ Imam, Su ¸u 46000, Kahramanmaras¸, Turkey E-mail address: [email protected]