Some Estimates for Generalized Riemann-Liouville Fractional Integrals of Exponentially Convex Functions and Their Applications

In the present paper, we investigate some Hermite-Hadamard ( HH ) inequalities related to generalized Riemann-Liouville fractional integral ( GRLFI ) via exponentially convex functions. We also show the fundamental identity for GRLFI having the first order derivative of a given exponentially convex function. Monotonicity and exponentially convexity of functions are used with some traditional and forthright inequalities. In the application part, we give examples and new inequalities for the special means.

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NEW INEQUALITIES OF OSTROWSKI TYPE FOR CO-ORDINATED CONVEX FUNCTIONS VIA GENERALIZED FRACTIONAL INTEGRALS

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi2004899a ◽

2021 ◽

pp. 899

Author(s):

Muhammad Aamir Ali ◽

Hüseyin Budak ◽

Zhiyue Zhang

Keyword(s):

Convex Function ◽

Convex Functions ◽

Fractional Integral ◽

Fractional Integrals ◽

Special Cases ◽

Generalized Fractional Integral ◽

New Inequalities

In this paper, we establish new inequalities of Ostrowski type for co-ordinated convex function by using generalized fractional integral. We also discuss some special cases of our established results.

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Fractional Ostrowski type inequalities for differentiable harmonically convex functions

AIMS Mathematics ◽

10.3934/math.2022217 ◽

2021 ◽

Vol 7 (3) ◽

pp. 3939-3958

Author(s):

Thanin Sitthiwirattham ◽

◽

Muhammad Aamir Ali ◽

Hüseyin Budak ◽

Sotiris K. Ntouyas ◽

...

Keyword(s):

Convex Functions ◽

Fractional Integrals ◽

Real Numbers ◽

Special Means ◽

Special Cases ◽

Harmonically Convex Functions ◽

New Inequalities

<abstract><p>In this paper, we prove some new Ostrowski type inequalities for differentiable harmonically convex functions using generalized fractional integrals. Since we are using generalized fractional integrals to establish these inequalities, therefore we obtain some new inequalities of Ostrowski type for Riemann-Liouville fractional integrals and $ k $-Riemann-Liouville fractional integrals in special cases. Finally, we give some applications to special means of real numbers for newly established inequalities.</p></abstract>

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Hermite–Hadamard-Type Inequalities for Convex Functions via the Fractional Integrals with Exponential Kernel

Mathematics ◽

10.3390/math7090845 ◽

2019 ◽

Vol 7 (9) ◽

pp. 845 ◽

Cited By ~ 5

Author(s):

Xia Wu ◽

JinRong Wang ◽

and Jialu Zhang

Keyword(s):

Real Number ◽

Convex Functions ◽

Fractional Integral ◽

Second Order ◽

Fractional Integrals ◽

Exponential Kernel ◽

Special Means ◽

Integral Identities

In this paper, we establish three fundamental integral identities by the first- and second-order derivatives for a given function via the fractional integrals with exponential kernel. With the help of these new fractional integral identities, we introduce a few interesting Hermite–Hadamard-type inequalities involving left-sided and right-sided fractional integrals with exponential kernels for convex functions. Finally, some applications to special means of real number are presented.

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On Strongly Convex Functions via Caputo–Fabrizio-Type Fractional Integral and Some Applications

Journal of Mathematics ◽

10.1155/2021/6625597 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

Qi Li ◽

Muhammad Shoaib Saleem ◽

Peiyu Yan ◽

Muhammad Sajid Zahoor ◽

Muhammad Imran

Keyword(s):

Integral Operator ◽

Fractional Calculus ◽

Convex Functions ◽

Fractional Integral ◽

Optimization Problems ◽

Fractional Integral Operator ◽

Strongly Convex Functions ◽

Strongly Convex ◽

Special Means ◽

New Inequalities

The theory of convex functions plays an important role in the study of optimization problems. The fractional calculus has been found the best to model physical and engineering processes. The aim of this paper is to study some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator. In this paper, we present Hermite–Hadamard-type inequalities for strongly convex functions via the Caputo–Fabrizio fractional integral operator. Some new inequalities of strongly convex functions involving the Caputo–Fabrizio fractional integral operator are also presented. Moreover, we present some applications of the proposed inequalities to special means.

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On Hadamard Type Fractional Inequalities for Riemann–Liouville Integrals via a Generalized Convexity

Fractal and Fractional ◽

10.3390/fractalfract6010028 ◽

2022 ◽

Vol 6 (1) ◽

pp. 28

Author(s):

Tao Yan ◽

Ghulam Farid ◽

Hafsa Yasmeen ◽

Chahn Yong Jung

Keyword(s):

Convex Function ◽

Convex Functions ◽

Fractional Integral ◽

Generalized Convexity ◽

Monotone Function ◽

Integral Inequalities ◽

Fractional Integrals ◽

Recent Past ◽

Hadamard Inequality

In the literature of mathematical inequalities, convex functions of different kinds are used for the extension of classical Hadamard inequality. Fractional integral versions of the Hadamard inequality are also studied extensively by applying Riemann–Liouville fractional integrals. In this article, we define (α,h−m)-convex function with respect to a strictly monotone function that unifies several types of convexities defined in recent past. We establish fractional integral inequalities for this generalized convexity via Riemann–Liouville fractional integrals. The outcomes of this work contain compact formulas for fractional integral inequalities which generate results for different kinds of convex functions.

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On Caputo–Fabrizio Fractional Integral Inequalities of Hermite–Hadamard Type for Modified h -Convex Functions

Journal of Mathematics ◽

10.1155/2020/8829140 ◽

2020 ◽

Vol 2020 ◽

pp. 1-17

Author(s):

Xiaobin Wang ◽

Muhammad Shoaib Saleem ◽

Kiran Naseem Aslam ◽

Xingxing Wu ◽

Tong Zhou

Keyword(s):

Integral Operator ◽

Fractional Calculus ◽

Convex Function ◽

Convex Functions ◽

Fractional Derivatives ◽

Fractional Integral ◽

Applied Mathematics ◽

Fractional Integral Operator ◽

Real Numbers ◽

Special Means

The theory of convex functions plays an important role in engineering and applied mathematics. The Caputo–Fabrizio fractional derivatives are one of the important notions of fractional calculus. The aim of this paper is to present some properties of Caputo–Fabrizio fractional integral operator in the setting of h -convex function. We present some new Caputo–Fabrizio fractional estimates from Hermite–Hadamard-type inequalities. The results of this paper can be considered as the generalization and extension of many existing results of inequalities and convex functions. Moreover, we also present some application of our results to special means of real numbers.

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On New Inequalities via Riemann-Liouville Fractional Integration

Abstract and Applied Analysis ◽

10.1155/2012/428983 ◽

2012 ◽

Vol 2012 ◽

pp. 1-10 ◽

Cited By ~ 24

Author(s):

Mehmet Zeki Sarikaya ◽

Hasan Ogunmez

Keyword(s):

Convex Functions ◽

Fractional Integral ◽

Fractional Integration ◽

Integral Inequalities ◽

Fractional Integrals ◽

New Inequalities

We extend the Montgomery identities for the Riemann-Liouville fractional integrals. We also use these Montgomery identities to establish some new integral inequalities. Finally, we develop some integral inequalities for the fractional integral using differentiable convex functions.

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New Conformable Fractional Integral Inequalities of Hermite–Hadamard Type for Convex Functions

Symmetry ◽

10.3390/sym11020263 ◽

2019 ◽

Vol 11 (2) ◽

pp. 263 ◽

Cited By ~ 10

Author(s):

Pshtiwan Mohammed ◽

Faraidun Hamasalh

Keyword(s):

Convex Functions ◽

Fractional Integral ◽

Integral Inequalities ◽

Fractional Integrals ◽

Conformable Fractional Integral ◽

New Inequalities

In this work, we established new inequalities of Hermite–Hadamard type for convex functions via conformable fractional integrals. Through the conformable fractional integral inequalities, we found some new inequalities of Hermite–Hadamard type for convex functions in the form of classical integrals.

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Inequalities Pertaining Fractional Approach through Exponentially Convex Functions

Fractal and Fractional ◽

10.3390/fractalfract3030037 ◽

2019 ◽

Vol 3 (3) ◽

pp. 37 ◽

Cited By ~ 7

Author(s):

Saima Rashid ◽

Muhammad Aslam Noor ◽

Khalida Inayat Noor

Keyword(s):

Convex Function ◽

Convex Functions ◽

Fractional Integrals ◽

Exponentially Convex Functions

In this article, certain Hermite-Hadamard-type inequalities are proven for an exponentially-convex function via Riemann-Liouville fractional integrals that generalize Hermite-Hadamard-type inequalities. These results have some relationships with the Hermite-Hadamard-type inequalities and related inequalities via Riemann-Liouville fractional integrals.

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Boundedness of Fractional Integral Operators Containing Mittag-Leffler Function via Exponentially s-Convex Functions

Journal of Mathematics ◽

10.1155/2020/3584105 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Gang Hong ◽

G. Farid ◽

Waqas Nazeer ◽

S. B. Akbar ◽

J. Pečarić ◽

...

Keyword(s):

Convex Functions ◽

Fractional Integral ◽

Symmetric Functions ◽

Integral Operators ◽

Arbitrary Point ◽

Fractional Integrals ◽

Hadamard Inequality ◽

Operator Inequalities ◽

Fractional Integral Operators ◽

Exponentially Convex Functions

The main objective of this paper is to obtain the fractional integral operator inequalities which provide bounds of the sum of these operators at an arbitrary point. These inequalities are derived for s-exponentially convex functions. Furthermore, a Hadamard inequality is obtained for fractional integrals by using exponentially symmetric functions. The results of this paper contain several such consequences for known fractional integrals and functions which are convex, exponentially convex, and s-convex.

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