Hermite-Hadamard type inequalities for F-convex function involving fractional integrals.

In this study, the family F and F-convex function are given with its properties. In view of this, we establish some new inequalities of Hermite-Hadamard type for differentiable function. Moreover, we establish some trapezoid type inequalities for functions whose second derivatives in absolute values are F-convex. We also show that through the notion of F-convex we can find some new Hermite-Hadamard type and trapezoid type inequalities for the Riemann-Liouville fractional integrals and classical integrals.

Introduction

A function is said to be convex on the interval I, if for all and it satisfies the following inequality: Convex functions play an important role in the field of integral inequalities. For convex functions, many equalities and inequalities have been established, but one of the most important ones is the Hermite–Hadamard’ integral inequality, which is defined as follows [1]:

Let be a convex function with and . Then the Hermite–Hadamard inequality is given by In recent years, a number of mathematicians have devoted their efforts to generalizing, refining, counterparting, and extending the Hermite–Hadamard inequality (2) for different classes of convex functions and mappings. The Hermite–Hadamard inequality (2) is established for the classical integral, fractional integrals, conformable fractional integrals and most recently for generalized fractional integrals; see for details and applications [2-8] and the references therein.

The concepts of classical convex functions have been extended and generalized in several directions, such as quasi-convex [9], pseudo-convex [10], MT-convex [11] strongly convex [12], ϵ-convex [13], s-convex [14], h-convex [15], and -preinvex [16]. Recently, Samet [17] has defined a new concept of convexity that depends on a certain function satisfying some axioms, generalizing different types of convexity, including ϵ-convex functions, α-convex functions, h-convex functions, and so on, as stated in the next section.

Review of the family of

We address the family of of mappings satisfying the following axioms:

If , , then, for every , we have

For every , and , we have where is a function that depends on , and it is nondecreasing with respect to the first variable.

For any , , we have where is a constant that depends only on w.

Definition 2.1

Let , , , be a given function. We say that f is a convex function with respect to some (or F-convex function) iff

Remark 1

Suppose that with .

Let be an ε-convex function, that is [18], Define the functions by and by For it will be seen that and that is, f is an F-convex function. Particularly, taking we show that if f is a convex function then f is an F-convex function with respect to F defined above.

Let be -preinvex function according to φ and bifunction η, , , that is [16], Define the functions by and by For , it will be seen that and that is f is an F-convex function.

Let be a given function which is not identical to 0, where I is an interval in such that . Let be an h-convex function, that is, Define the functions by and by For , it will be seen that and that is f is an F-convex function.

Recently Samet [17] established some integral inequalities of Hermite–Hadamard type via F-convex functions.

Theorem 1

([17, Theorem 3.1])

Let , , , be an F-convex function, for some . Suppose that . Then

Theorem 2

([17, Theorem 3.4])

Let be a differentiable mapping on , , . Suppose that Then

is F-convex on , for some

the function belongs to , where .

Theorem 3

([17, Theorem 3.5])

Let be a differentiable mapping on , , and let . Suppose that is F-convex on , for some and . Then where

As consequences of the above theorems, the author obtained some integral inequalities for ε-convexity, α-convexity, and h-convexity.

Theorem 4

([17, Corollary 4.3])

Let be a differentiable mapping on , , . Suppose that the function is ε-convex on , . Then

Theorem 5

([17, Corollary 4.9])

Let be a differentiable mapping on , , . Suppose that the function is α-convex on , . Then

Theorem 6

([17, Corollary 4.14])

Let be a differentiable mapping on , , . Suppose that the function is h-convex on . Then

For more recent results on integral inequalities of Hermite–Hadamard type concerning the F-convex functions, we refer the interested reader to [19] and the references therein.

In the sequel, we recall the concepts of the left-sided and right-sided Riemann- Liouville fractional integrals of the order .

Definition 2.2

([20])

Suppose that . The left and right Riemann–Liouville fractional integrals denoted by and of order are defined by and respectively, where is the gamma function defined by and .

In [21], authors established the following Hermite–Hadamard type inequalities for F-convex functions involving a Riemann–Liouville fractional:

Theorem 7

Let be an interval, be a differentiable mapping on , , . If f is F-convex on , for some , then we have where .

Theorem 8

Let be an interval, be a differentiable mapping on , , . If f is F-convex on , for some and the function belongs to , where . Then we have the inequality

The following definitions will be useful for this study [20].

Definition 2.3

The Euler beta function is defined as follows: The incomplete beta function is defined by

Note that, for , the incomplete beta function reduces to the Euler beta function.

Also, the following three lemmas are important to obtain our main results.

Lemma 1

([22, Lemma 4])

Let be a once differentiable mappings on with , . If , then the following equality for the fractional integral holds:

Lemma 2

([16, Lemma 5])

Let be a once differentiable mappings on with , . If , then the following equality for the fractional integral holds:

Lemma 3

([22])

For , we have

In this study, using the -preinvexity of the function, we establish new inequalities of Hermite–Hadamard type for differentiable function and some trapezoid type inequalities for function whose second derivatives absolutely values are F-convex.

Hermite–Hadamard type inequalities for differentiable functions

In this section, we establish some inequalities of Hermite–Hadamard type for F-convex functions in fractional integral forms.

Theorem 9

Let be an open invex set with respect to bifunction , where . Let be a differentiable mapping. Suppose that is measurable, decreasing, -preinvex function on I, and F-convex on , for some and the function belongs to , where . Then

Proof

Since is F-convex, we have Multiplying this inequality by and using axiom (A3), we have Integrating over and using axiom (A2), we get

But from Lemma 1 we have Because is nondecreasing with respect to the first variable so that This proves (10). □

Remark 2

If we choose and in Theorem 9, we get

Corollary 1

Under the assumptions of Theorem 9, if is ε-convex, then we have

Using (5) with , we find From (4) with , we have for . Hence, by Theorem 9, we have This completes the proof. □

Remark 3

In Corollary 1, if we choose

and , we get

, , and , we get which is given by [18].

Corollary 2

Under the assumptions of Theorem 9, if is -preinvex, then we have

Using (7) with , we have for . Hence, by Theorem 9, we have This leads to Thus, the proof is done. □

Remark 4

In Corollary 2, if we choose

and , we get

, , and , we get

Corollary 3

Under the assumptions of Theorem 9, if is h-convex, then we have

Using (9) with , we have for . So, by Theorem 9, we have This leads to Thus, the proof is done. □

Theorem 10

Let be an open invex set with respect to bifunction , where . Let be a differentiable mapping. Suppose that is measurable, decreasing, -preinvex function on I, and F-convex on , for some and . Then where

Since is F-convex, we have With in (A2), we have Using Lemma 1 and the Hölder inequality, we get or, equivalently, Because is nondecreasing with respect to the first variable, we get Thus, the proof is completed. □

Remark 5

If we choose and in Theorem 10, we get

Corollary 4

Under the assumptions of Theorem 10, if is ε-convex, we have

Using (5) with , we have From (4) with , we have for . Hence, by Theorem 10, we have This leads to or, equivalently, This completes the proof. □

Remark 6

In Corollary 4, if we choose

and , we get

, , and , we get

Corollary 5

Under the assumptions of Theorem 10. If is -preinvex, we have

Using (7) with , we have for . So, by Theorem 10, we have This leads to Thus, the proof is done. □

Remark 7

In Corollary 5, if we choose

and , we get

, , and , we get

Corollary 6

Under the assumptions of Theorem 10. If is h-convex, we have

From (9) with , we have for . So, by Theorem 9, we have that is, This completes the proof. □

Theorem 11

Let be an open invex set with respect to bifunction , where . Let be a differentiable mapping. Suppose that is measurable, decreasing, -preinvex function on I, and F-convex on , for some and . Then where for .

Since is F-convex, we have Using (A3) with , we obtain Integrating over and using axiom (A2), we obtain Using Lemma 1 and the power mean inequality, we get or, equivalently, Because is nondecreasing with respect to the first variable, we find This completes the proof. □

Remark 8

If we choose and in Theorem 11, we get

Corollary 7

Under the assumptions of Theorem 11, if is ε-convex, we have

Using (5) with , we get From (4) with , we get for . Hence, by Theorem 10, we have This implies that This completes the proof. □

Remark 9

In Corollary 7, if we choose

and , we get

, , and , we get

Corollary 8

Under the assumptions of Theorem 11. If is -preinvex, we have

Using (7) with , we have for . Now, by Theorem 11, we have This leads to Thus, the proof is done. □

Remark 10

In Corollary 8, if we choose

and , we get

, , and , we get

Corollary 9

Under the assumptions of Theorem 11. If is h-convex, we have

From (9) with , we have for . So, by Theorem 11, we have that is, This completes the proof. □

Trapezoid type inequalities for twice differentiable functions

In this section, we establish some trapezoid type inequalities for functions whose second derivatives absolutely values are

Theorem 12

Let be a differentiable mapping and is measurable, decreasing, -preinvex function on for , and . Suppose that F-convex on , for some and the function belongs to , where . Then

Since is F-convex, we can see that Multiplying this inequality by and using axiom (A3), we have Integrating over and using axiom (A2), we get

Using Lemma 2, we have Because is nondecreasing with respect to the first variable so that This completes the proof. □

Remark 11

By taking and in Theorem 12, we obtain

Corollary 10

Under the assumptions of Theorem 12, if is ε-convex, then

Using (5) with , we find With , Eq. (4) gives for . Hence, by Theorem 12, we have This completes the proof. □

Remark 12

In Corollary 10, if we take

and , we get

, , and , we get

Corollary 11

Under the assumptions of Theorem 12, if is -preinvex, then

Using (7) with , we have for . Hence, by Theorem 12, we get This leads to Thus, the proof is completed. □

Remark 13

In Corollary 11, if we choose

and , we get

, , and , we get

Corollary 12

Under the assumptions of Theorem 12, if is h-convex, then we have

Using (9) with , we obtain for , so Theorem 12 implies that which can be written as Thus, the proof is done. □

Theorem 13

Let be a differentiable mapping and is measurable, decreasing, -preinvex function on for and . Suppose that is F-convex on , for some and , . Then we have where

Since is F-convex, we have Using (A2) with , we have Using Lemma 2, Lemma 3 and the Hölder inequality, we get or, equivalently, Because is nondecreasing with respect to the first variable, we get Thus, the proof is completed. □

Remark 14

If we choose and in Theorem 13, we get

Corollary 13

Under the assumptions of Theorem 13, if is ε-convex, then

Using (12), (13), by Theorem 13, we have This leads to This completes the proof. □

Remark 15

In Corollary 13, if we choose

and , we get

, , and , we get

Corollary 14

Under the assumptions of Theorem 13. If is -preinvex, we have

Using (7), (14), by Theorem 13, we have that is, This proves Corollary 14. □

Remark 16

In Corollary 14, if we choose

and , we get

, , and , we get

Corollary 15

Under the assumptions of Theorem 13. If is h-convex, then

Using (9) and by Theorem 13, it can be proved easily. It is omitted. □

Theorem 14

Let be a differentiable mapping and is measurable, decreasing, -preinvex function on for and . Suppose that is F-convex on , for some and , . Then we have where for .

Since is F-convex, we have Using (A3) with , we obtain Integrating over and using axiom (A2), we obtain Using Lemma 2 and the power mean inequality, we get or, equivalently, Since is nondecreasing with respect to the first variable, we have This completes the proof. □

Remark 17

Taking and in Theorem 14, we get

Corollary 16

Under the assumptions of Theorem 14, if is ε-convex, we have

Using (17), (18) and by Theorem 13, we obtain This completes the proof. □

Remark 18

In Corollary 16, if we choose

and , we get

, , and , we get

Corollary 17

Under the assumptions of Theorem 14. If is -preinvex, then

Using (19), by Theorem 14, we have This leads to This ends the proof. □

Remark 19

In Corollary 17, if we choose

and , we get

, , and , we get

Corollary 18

Under the assumptions of Theorem 14. If is h-convex, we have

Using (9), by Theorem 14, it can be proved easily. It is omitted. □

Conclusion

In the present paper, using the notion of F and F-convex function (see [17]), we construct some new inequalities of Hermite–Hadamard type for differentiable function via Riemann–Liouville fractional integral. We also established some trapezoid type inequalities for a function of whose second derivatives absolutely values are F-convex. Moreover, we obtained some new inequalities of Hermite–Hadamard type for Riemann–Liouville fractional integrals and via classical integrals. The results presented in this paper would provide generalizations and extension of those given in earlier work.