Turkish Journal of Analysis and Number Theory
Volume 10, 2022 - Issue 1
Website: http://www.sciepub.com/journal/tjant

ISSN(Print): 2333-1100
ISSN(Online): 2333-1232

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Research Article

Open Access Peer-reviewed

Muhammad Muddassar^{ }, Sajida Batool, Tahira Jabeen

Received June 20, 2022; Revised July 25, 2022; Accepted August 03, 2022

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.

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

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

**De****finition 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:

**Theo****rem 21** ^{ 10} *It is assume that ** is a positive mapping, here ** with ** and ** when*

**Theo****rem 2****2** ^{ 10} *It is assume that ** is a differentiable function on ** where ** such that ** and ** Then*

(16) |

*Let ** is a filled subset of ** and let *

**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}.

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.

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.

All authors declare that they have no competing interests.

All authors participated to formulate this full length article.

No funding was received for this work.

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

[1] | Mitrinovic, D. S., Lackovic, I. B., (1985), Hermite and convexity, Aequat. Math. 28, pp.229-232. | ||

In article | View Article | ||

[2] | Ozdemir, M. E., Yildiz C., Akdemir, A. O., Set, E., (2013), On some inequalities for s-convex functions and applications, J. Ineq. Appl., Article ID 333. | ||

In article | View Article | ||

[3] | Set, E., Ozdemir, M. E., Sarikaya, M. Z., (2010), Inequalities of Hermite-Hadamard’s type for functions whose derivatives absolute values are m-convex, AIP Conference Proceedings, 1309(1) , pp.861-873. | ||

In article | View Article | ||

[4] | Yin, H. P., & Wang, J. Y. (2018). Some Integral Inequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions. Miskolc Mathematical Notes, 19(1), 699-705. | ||

In article | View Article | ||

[5] | Qi, F., Li, W-H., (2016), Integral representation and properties of some functions involving the logarithmic function, Filomat 30 no.7, pp.1659-1674. | ||

In article | View Article | ||

[6] | Mubeen, S., Iqbal, S., (2016), Gruss type integral inequalities for generalization Riemannliouville k-fractional integrals, Jour. Ineq. Appl. Paper No. 109, 13 pp. | ||

In article | View Article | ||

[7] | Cotez, M. V., Kashuri, A., Heranndez, J.E., (2020), Trapezium type inequalities for Raina’s fractional integrals operator using generalized convex functions, Symmetry, 12, pp.1206-1034. | ||

In article | View Article | ||

[8] | Kermausuor S., Nwaeze, E.R., (2021), New integral inequalities of Hermite-Hadamard type via the Katugampola fractional integrals for Strongly η-Quasiconvex functions, The Jour. Anal., 29: 633-647. | ||

In article | View Article | ||

[9] | Muddassar, M., Dragomir, S. S., Hussain, Z., (2021), Generalization of k-fractional conformable integral inequalities by the way of set-valued non-convex functions (Accepted, Arab Jour. Math. Sc.). | ||

In article | |||

[10] | Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N., (2013), Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57, pp.2403-2407. | ||

In article | View Article | ||

[11] | Aslani, S. M., Delavar, R.,Vaezpour, S.M., (2018), Inequalities of Fejer type related to generalized convex functions with applications, Int. J. Anal. Appl., 16(1), pp. 38-49. | ||

In article | |||

[12] | Chen, H., Katugampola, U.N., (2017), Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for generalized fractional integral, Journal of Mathematical Analysis and Applications 446(2), pp. 1247-1291. | ||

In article | View Article | ||

[13] | Delavar, M.R., La Sen, M. De, (2016), Some generalizations of Hermite-Hadamard type inequalities, Springerplus, 5: 1661, pp.1-9. | ||

In article | View Article PubMed | ||

[14] | Omotoyinbo, O., Mogbademu, A., (2017), Some convex functions for Hermite-Hadamard integral inequalities, Scientia Magna Vol. 12 (1), 14-22. | ||

In article | View Article | ||

[15] | Set, E., Sarikaya, M. Z., Ozdemir, M. E., (2015), On new inequalities of Hermite-Hadamard-Fejer type for convex functions via fractional integrals, Appl. Math. Comput. 259(C), pp.875-881. | ||

In article | View Article | ||

[16] | Set, E., Sarikaya, M. Z., Ozdemir, M. E., Yildirim, H., (2014), The Hermite-Hadamard’s inequality for some convex functions via fractional integrals and related results, J. Appl. Math. Stat. Inf. (JAMSI). 10(2), pp. 69-83. | ||

In article | View Article | ||

[17] | SET, E., CHOI, J., & GÖZPINAR, A. (2018). Hermite-Hadamard type inequalities for new conformable fractional integral operator. ResearchGate Preprint 2018. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2022 Muhammad Muddassar, Sajida Batool and Tahira Jabeen

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Muhammad Muddassar, Sajida Batool, Tahira Jabeen. Trapezoidal Error Approximation by the Way of Integral Inequalities for Katugampola Fractional Integral Operator. *Turkish Journal of Analysis and Number Theory*. Vol. 10, No. 1, 2022, pp 15-20. http://pubs.sciepub.com/tjant/10/1/4

Muddassar, Muhammad, Sajida Batool, and Tahira Jabeen. "Trapezoidal Error Approximation by the Way of Integral Inequalities for Katugampola Fractional Integral Operator." *Turkish Journal of Analysis and Number Theory* 10.1 (2022): 15-20.

Muddassar, M. , Batool, S. , & Jabeen, T. (2022). Trapezoidal Error Approximation by the Way of Integral Inequalities for Katugampola Fractional Integral Operator. *Turkish Journal of Analysis and Number Theory*, *10*(1), 15-20.

Muddassar, Muhammad, Sajida Batool, and Tahira Jabeen. "Trapezoidal Error Approximation by the Way of Integral Inequalities for Katugampola Fractional Integral Operator." *Turkish Journal of Analysis and Number Theory* 10, no. 1 (2022): 15-20.

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[1] | Mitrinovic, D. S., Lackovic, I. B., (1985), Hermite and convexity, Aequat. Math. 28, pp.229-232. | ||

In article | View Article | ||

[2] | Ozdemir, M. E., Yildiz C., Akdemir, A. O., Set, E., (2013), On some inequalities for s-convex functions and applications, J. Ineq. Appl., Article ID 333. | ||

In article | View Article | ||

[3] | Set, E., Ozdemir, M. E., Sarikaya, M. Z., (2010), Inequalities of Hermite-Hadamard’s type for functions whose derivatives absolute values are m-convex, AIP Conference Proceedings, 1309(1) , pp.861-873. | ||

In article | View Article | ||

[4] | Yin, H. P., & Wang, J. Y. (2018). Some Integral Inequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions. Miskolc Mathematical Notes, 19(1), 699-705. | ||

In article | View Article | ||

[5] | Qi, F., Li, W-H., (2016), Integral representation and properties of some functions involving the logarithmic function, Filomat 30 no.7, pp.1659-1674. | ||

In article | View Article | ||

[6] | Mubeen, S., Iqbal, S., (2016), Gruss type integral inequalities for generalization Riemannliouville k-fractional integrals, Jour. Ineq. Appl. Paper No. 109, 13 pp. | ||

In article | View Article | ||

[7] | Cotez, M. V., Kashuri, A., Heranndez, J.E., (2020), Trapezium type inequalities for Raina’s fractional integrals operator using generalized convex functions, Symmetry, 12, pp.1206-1034. | ||

In article | View Article | ||

[8] | Kermausuor S., Nwaeze, E.R., (2021), New integral inequalities of Hermite-Hadamard type via the Katugampola fractional integrals for Strongly η-Quasiconvex functions, The Jour. Anal., 29: 633-647. | ||

In article | View Article | ||

[9] | Muddassar, M., Dragomir, S. S., Hussain, Z., (2021), Generalization of k-fractional conformable integral inequalities by the way of set-valued non-convex functions (Accepted, Arab Jour. Math. Sc.). | ||

In article | |||

[10] | Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N., (2013), Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57, pp.2403-2407. | ||

In article | View Article | ||

[11] | Aslani, S. M., Delavar, R.,Vaezpour, S.M., (2018), Inequalities of Fejer type related to generalized convex functions with applications, Int. J. Anal. Appl., 16(1), pp. 38-49. | ||

In article | |||

[12] | Chen, H., Katugampola, U.N., (2017), Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for generalized fractional integral, Journal of Mathematical Analysis and Applications 446(2), pp. 1247-1291. | ||

In article | View Article | ||

[13] | Delavar, M.R., La Sen, M. De, (2016), Some generalizations of Hermite-Hadamard type inequalities, Springerplus, 5: 1661, pp.1-9. | ||

In article | View Article PubMed | ||

[14] | Omotoyinbo, O., Mogbademu, A., (2017), Some convex functions for Hermite-Hadamard integral inequalities, Scientia Magna Vol. 12 (1), 14-22. | ||

In article | View Article | ||

[15] | Set, E., Sarikaya, M. Z., Ozdemir, M. E., (2015), On new inequalities of Hermite-Hadamard-Fejer type for convex functions via fractional integrals, Appl. Math. Comput. 259(C), pp.875-881. | ||

In article | View Article | ||

[16] | Set, E., Sarikaya, M. Z., Ozdemir, M. E., Yildirim, H., (2014), The Hermite-Hadamard’s inequality for some convex functions via fractional integrals and related results, J. Appl. Math. Stat. Inf. (JAMSI). 10(2), pp. 69-83. | ||

In article | View Article | ||

[17] | SET, E., CHOI, J., & GÖZPINAR, A. (2018). Hermite-Hadamard type inequalities for new conformable fractional integral operator. ResearchGate Preprint 2018. | ||

In article | View Article | ||