Abstract

In the paper under consideration, we will discuss integral inequalities by the way of fractional integral operators for generalized convex functions. For fractional integral operators, a new identity is established and some results of Hermite-Hadamard type inequalities are achieved. This fractional operator have been used to derive a new generalization of the Hermite-Hadamard inequality. In the previous papers the author used the definition of k-fractional conformable integral inequalities and we used the definition of katugampola fractional integrals in this paper. In this paper, we have given a best appropriate definition for fractional integral operators.

1. Introduction

The fractional integral inequalities theory performs an essential role in science such as mathematics, physics and engineering of the type of Hermite-Hadamard inequality is another well-known inequality for generalized convex functions. These inequalities indicate that if is a convex function on the real numbers of boundaries and as when

(1)

is called Hermite-Hadamard inequality.

If is concave, both inequalities in 1 hold in the opposite direction.

In the literature some extensions, generalization and variant can be found. For detail information, see ^{1, 2, 3} and ⁴ (and the references given in). The Riemann-Liouville integrals and of order can be defined respectively.

(2)

(3)

where and the classical Euler gamma function is ⁵. The Riemann-Liouville k-fractional integral are given respectively, in ⁶.

(4)

(5)

For

2. Preliminaries

This part includes basic definitions as well as certain outcomes that they would be used in present work which is in progress. The following is the standard definition of -convex function.

Definition 1 ⁷ Let . If a mapping as follows, the set I is said to be a -convex,

(6)

for all and

Definition 2 ⁷ It is said to be -convex a real valued function g defined on -convex set I if,

(7)

The mapping g is called -concave iff is -convex. It can be seen that each convex mapping (function) is -convex but in general the converse does not hold. Generalized convexity and its application like h-convexity, η-convexity, -convexity and others, are discuss in the following references ^{12, 13, 14, 15, 16}. And recent few more results ^{11, 17} and further references there in.

Definition 3 ⁸ The both sided Riemann-Liouville fractional integral of power of g are represented by

(8)

and

(9)

here k is positive and is called the k-gamma function is as follows,

(10)

with the properties that

Let which is finite interval. Further both sided katugampola fractional integrals of order which is positive of are as follows,

(11)

and

(12)

with and if the integrals exist then is positive.

Remark 1: The katugampola fractional integral operators are well-defined on

Definition 5 ⁹ The generalized -fractional conformable integral operations are respectively.

(13)

and

(14)

where

Now we recall the Hermite-Hadamard type inequalities concerning the Riemann-Liouville fractional integral from ¹⁰ are as follow:

Theorem 21 ¹⁰ It is assume that is a positive mapping, here with and when

Theorem 22 ¹⁰ It is assume that is a differentiable function on where such that and Then

(16)

Let is a filled subset of and let

3. Main Results

Lemma 31 It is assume that is a differentiable mapping where as follows and and integral is of the type of katugampola fractional, then we have

(17)

Proof Let

(18)

(19)

Now integrating , we obtain

(20)

(21)

in similar way

(22)

(21) and (22) together imply (17). This completes the proof.

Remark 2: when we have the integral of the type of -fractional conformable in Lemma 31, we get Lemma 1 of ⁹.

Let be such that and , if is invex on [c, d], then for the integral operator of the type of katugampola fractional gives

(23)

for

Proof As denote invex function on [c, d], we obtain

(24)

Letting and gives

(25)

Multiplying on both side of (25) by and integrating with respect to z over [0,1] leads to

(26)

(27)

(28)

(29)

(30)

From

(31)

it follow that

(32)

the above can be written as the left side of inequality (23) making use of the invexity, comes at

(33)

and

(34)

adding the above two inequalities yields

(35)

Now integrating on both sides (35) with respect to over [0,1], and interchanging the parameters in the second integration on the right side, we get

(36)

the above can be rewritten as the right hand side of inequality (23). This completes the result in theorem 32.

Remark 3: By utilizing the definition of -fractional conformable in Theorem 32, Gives Theorem 3 of ⁹.

Theorem 33 Let be a differentiable mapping as follows and and . if is invex on [c, d], then for the integral operator of the type of katugampola fractional gives

(37)

for

Proof By lemma 31 and in the invexity of , we have

(38)

(39)

(40)

(41)

changing variable by and results in

(42)

(43)

(44)

(45)

substituting the equalities (42), (43), (44) and (45) into the equality (41) leads to (37). This completes the result in theorem 33.

Remark 4: By utilizing the definition of -fractional conformable in theorem 33, becomes Theorem 4 of ⁹.

4. Application: Trapezoidal Error Approximation

Let be the partition

of the interval [c, d] and take into account the quadrature formula

(46)

Where

(47)

The associated approximation error is represented by .

Proposition 41 Let is a differentiable mapping on such that when as and is a convex on [c, d]. For every partition of [c, d] the trapezoidal error approximation fulfills,

(48)

where

Proof Theorem 33 is applied to the subinterval of the division of [c, d] with , and put we have

(49)

Taking the sum over k from to then we have

(50)

(51)

it gives,

(52)

(53)

By combining (52) and (53) we get (48). This gives the error approximation for Trapezoidal type inequalities.

5. Concluding Remarks

We have discovered certain solutions that we get from the belonging of k-fractional conformable integral inequalities and the use of katugampola fractional integrals in this exploration of the objective. Similarly, equivalent results for invex functions were established, generalising previous results in literature employing Riemann-liouville fractional integrals. Our findings are related to the Hermite-Hadamard integral inequality, which is a well-known integral inequality.

Competing Interest

All authors declare that they have no competing interests.

Authors Contribution

All authors participated to formulate this full length article.

Funding Information

No funding was received for this work.

Data Availability

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References