scholarly journals New Ostrowski-Type Fractional Integral Inequalities via Generalized Exponential-Type Convex Functions and Applications

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1429
Author(s):  
Soubhagya Kumar Sahoo ◽  
Muhammad Tariq ◽  
Hijaz Ahmad ◽  
Jamshed Nasir ◽  
Hassen Aydi ◽  
...  
Keyword(s):  
Fractional Integral ◽  
Great Part ◽  
Real Life ◽  
Strong Relationship ◽  
Integral Inequalities ◽  
Positive Real ◽  
New Class ◽  
Life Problems ◽  
Special Cases ◽  
Mathematical Sciences

Recently, fractional calculus has been the center of attraction for researchers in mathematical sciences because of its basic definitions, properties and applications in tackling real-life problems. The main purpose of this article is to present some fractional integral inequalities of Ostrowski type for a new class of convex mapping. Specifically, n–polynomial exponentially s–convex via fractional operator are established. Additionally, we present a new Hermite–Hadamard fractional integral inequality. Some special cases of the results are discussed as well. Due to the nature of convexity theory, there exists a strong relationship between convexity and symmetry. When working on either of the concepts, it can be applied to the other one as well. Integral inequalities concerned with convexity have a lot of applications in various fields of mathematics in which symmetry has a great part to play. Finally, in applications, some new limits for special means of positive real numbers and midpoint formula are given. These new outcomes yield a few generalizations of the earlier outcomes already published in the literature.

Some new fractional integral inequalities for generalized relative semi-m-(r; h1, h2)-preinvex mappings via generalized Mittag-Leffler function

2019 ◽  
Vol 26 (1/2) ◽  
pp. 41-55 ◽  
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Invex Set ◽  
Special Means ◽  
Special Cases ◽  
Differentiable Mappings

The authors discover a new identity concerning differentiable mappings defined on m-invex set via fractional integrals. By using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized relative semi- m-(r;h1,h2)-preinvex mappings by involving generalized Mittag-Leffler function are presented. It is pointed out that some new special cases can be deduced from main results of the paper. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

scholarly journals Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1485
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Shengda Zeng ◽  
Artion Kashuri
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Applied Mathematics ◽  
Strong Relationship ◽  
Integral Inequalities ◽  
Mathematical Methods ◽  
Fractional Integral Operators ◽  
New Class ◽  
Definition Of

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of φ-Riemann–Liouville fractional integral operators and the recently defined class of convex functions, namely the σ˘-convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a σ˘-convex function with respect to an increasing function involving the φ-Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out.

scholarly journals Some Hermite–Hadamard-Type Fractional Integral Inequalities Involving Twice-Differentiable Mappings

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2209
Author(s):  
Soubhagya Kumar Sahoo ◽  
Muhammad Tariq ◽  
Hijaz Ahmad ◽  
Ayman A. Aly ◽  
Bassem F. Felemban ◽  
...  
Keyword(s):  
Applied Sciences ◽  
Focal Point ◽  
Real Life ◽  
Integral Inequalities ◽  
Mathematical Science ◽  
Life Problems ◽  
Special Cases ◽  
Wide Range ◽  
Fractional Analysis ◽  
Almost All

The theory of fractional analysis has been a focal point of fascination for scientists in mathematical science, given its essential definitions, properties, and applications in handling real-life problems. In the last few decades, many mathematicians have shown their considerable interest in the theory of fractional calculus and convexity due to their wide range of applications in almost all branches of applied sciences, especially in numerical analysis, physics, and engineering. The objective of this article is to establish Hermite-Hadamard type integral inequalities by employing the k-Riemann-Liouville fractional operator and its refinements, whose absolute values are twice-differentiable h-convex functions. Moreover, we also present some special cases of our presented results for different types of convexities. Moreover, we also study how q-digamma functions can be applied to address the newly investigated results. Mathematical integral inequalities of this class and the arrangements associated have applications in diverse domains in which symmetry presents a salient role.

scholarly journals Fractional integral inequalities for generalized-$$\mathbf{m }$$-$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$-convex mappings via an extended generalized Mittag–Leffler function

2019 ◽  
Vol 9 (2) ◽  
pp. 231-243
Author(s):  
George Anastassiou ◽  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Convex Mappings ◽  
Invex Set ◽  
Special Means ◽  
Special Cases

AbstractThe authors discover a new identity concerning differentiable mappings defined on $$\mathbf{m }$$ m -invex set via general fractional integrals. Using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized-$$\mathbf{m }$$ m -$$((h_{1}^{p},h_{2}^{q});(\eta _{1},\eta _{2}))$$ ( ( h 1 p , h 2 q ) ; ( η 1 , η 2 ) ) -convex mappings by involving an extended generalized Mittag–Leffler function are presented. It is pointed out that some new special cases can be deduced from main results. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

scholarly journals New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1753
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Omar Bazighifan ◽  
Georgia Irina Oros
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Type Inequality ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Convex Mappings ◽  
Special Cases ◽  
Harmonically Convex Functions

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

scholarly journals On Fractional Integral Inequalities Involving Hypergeometric Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
D. Baleanu ◽  
S. D. Purohit ◽  
Praveen Agarwal
Keyword(s):  
Hypergeometric Function ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Gauss Hypergeometric Function ◽  
Liouville Type ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Chebyshev Functional

Here we aim at establishing certain new fractional integral inequalities involving the Gauss hypergeometric function for synchronous functions which are related to the Chebyshev functional. Several special cases as fractional integral inequalities involving Saigo, Erdélyi-Kober, and Riemann-Liouville type fractional integral operators are presented in the concluding section. Further, we also consider their relevance with other related known results.

scholarly journals New Hermite-Hadamard-Fejér inequalities via k-fractional integrals for differentiable generalized nonconvex functions

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2549-2558
Author(s):  
Artion Kashuri ◽  
Themistocles Rassias
Keyword(s):  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Nonconvex Functions ◽  
New Class ◽  
Differentiable Functions ◽  
Special Cases

The authors discover a new interesting generalized identity concerning differentiable functions via k-fractional integrals. By using the obtained identity as an auxiliary result, some new estimates with respect to Hermite-Hadamard-Fej?r type inequalities via k-fractional integrals for a new class of function involving Raina?s function, the so-called generalized (h1, h2)-nonconvex are presented. These inequalities have some connections with known integral inequalities. Also, some new special cases are provided as well from main results.

scholarly journals New integral inequalities for strongly nonconvex functions involving Raina function

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2437-2456
Author(s):  
Artion Kashuri ◽  
Marcela Mihai ◽  
Muhammad Awan ◽  
Muhammad Noor ◽  
Khalida Noor
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Nonconvex Functions ◽  
Nonconvex Function ◽  
Special Cases ◽  
Type Integral ◽  
Generalized Fractional Integral ◽  
General Class Of Functions ◽  
Class Of Functions

In this paper, the authors defined a new general class of functions, the so-called strongly (h1,h2)-nonconvex function involving F??,?(?) (Raina function). Utilizing this, some Hermite-Hadamard type integral inequalities via generalized fractional integral operator are obtained. Some new results as a special cases are given as well.

scholarly journals Tempered Fractional Integral Inequalities for Convex Functions

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 500 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Special Cases ◽  
New Inequalities

Certain new inequalities for convex functions by utilizing the tempered fractional integral are established in this paper. We also established some new results by employing the connections between the tempered fractional integral with the (R-L) fractional integral. Several special cases of the main result are also presented. The obtained results are more in a general form as it reduced certain existing results of Dahmani (2012) and Liu et al. (2009) by employing some particular values of the parameters.

scholarly journals Integral inequalities involving generalized Erdélyi-Kober fractional integral operators

2016 ◽  
Vol 14 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Dumitru Baleanu ◽  
Sunil Dutt Purohit ◽  
Jyotindra C. Prajapati
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Weighted Version ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Chebyshev Functional

AbstractUsing the generalized Erdélyi-Kober fractional integrals, an attempt is made to establish certain new fractional integral inequalities, related to the weighted version of the Chebyshev functional. The results given earlier by Purohit and Raina (2013) and Dahmani et al. (2011) are special cases of results obtained in present paper.

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