scholarly journals Inequalities for Riemann–Liouville Fractional Integrals of Strongly s , m -Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Fuzhen Zhang ◽  
Ghulam Farid ◽  
Saira Bano Akbar
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Error Estimations

The results of this paper provide two Hadamard-type inequalities for strongly s , m -convex functions via Riemann–Liouville fractional integrals and error estimations of well-known fractional Hadamard inequalities. Their special cases are given and connected with the results of some published papers.

scholarly journals Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dafang Zhao ◽  
Muhammad Aamir Ali ◽  
Artion Kashuri ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Definition Of ◽  
The Given ◽  
Class Of Functions ◽  
Interval Valued ◽  
New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

scholarly journals Fractional Ostrowski type inequalities for differentiable harmonically convex functions

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>

scholarly journals Trapezium-Type Inequalities for an Extension of Riemann–Liouville Fractional Integrals Using Raina’s Special Function and Generalized Coordinate Convex Functions

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 117
Author(s):  
Miguel Vivas-Cortez ◽  
Artion Kashuri ◽  
Rozana Liko ◽  
Jorge Eliecer Hernández Hernández
Keyword(s):  
Convex Functions ◽  
Special Function ◽  
Fractional Integrals ◽  
Generalized Convex Functions ◽  
Hadamard Inequality ◽  
Trapezium Type ◽  
Several Variables ◽  
Special Cases ◽  
Interesting Identity ◽  
Generalized Coordinate

In this paper, the authors analyse and study some recent publications about integral inequalities related to generalized convex functions of several variables and the use of extended fractional integrals. In particular, they establish a new Hermite–Hadamard inequality for generalized coordinate ϕ-convex functions via an extension of the Riemann–Liouville fractional integral. Furthermore, an interesting identity for functions with two variables is obtained, and with the use of it, some new extensions of trapezium-type inequalities using Raina’s special function via generalized coordinate ϕ-convex functions are developed. Various special cases have been studied. At the end, a brief conclusion is given as well.

scholarly journals On Hermite-Hadamard type inequalities of coordinated (p1,h1)-(p2,h2)-convex functions via Katugampola fractional integrals

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4785-4802
Author(s):  
Muhammad Raees ◽  
Matloob Anwar
Keyword(s):  
Present Article ◽  
Convex Functions ◽  
Classical Case ◽  
Fractional Integrals ◽  
Special Cases

The present article establishes some results on Hermite-Hadamard type inequalities of coordinated (p1,h1)-(p2,h2)-convex functions by using Katugampola fractional integrals. The results are in generalized form and deduced for various special cases like coordinated pq-convex functions, coordinated (p,s)-convex functions, coordinated s-convex functions and classical case of coordinated convex functions.

scholarly journals Some integral inequalities for approximately h—convex functions and their applications

2021 ◽  
Vol 40 (2) ◽  
pp. 481-504
Author(s):  
Artion Kashuri ◽  
Muhammad Raees ◽  
Matloob Anwar
Keyword(s):  
Convex Function ◽  
Integral Inequality ◽  
Convex Functions ◽  
Integral Inequalities ◽  
Integral Type ◽  
Special Cases ◽  
Hölder’S Integral Inequality ◽  
Error Estimations

In this paper, by applying the new and improved form of Hölder’s integral inequality called Hölder—Íşcan integral inequality three inequalities of Hermite—Hadamard and Hadamard integral type for (h, d)—convex functions have been established. Various special cases including classes for instance, h—convex, s—convex function of Breckner and Godunova—Levin—Dragomir and strong versions of the aforementioned types of convex functions have been identified. Some applications to error estimations of presented results have been analyzed. At the end, a briefly conclusion is given.

scholarly journals Fractional Versions of Hadamard-Type Inequalities for Strongly Exponentially α , h − m -Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Shasha Li ◽  
Ghulam Farid ◽  
Atiq Ur Rehman ◽  
Hafsa Yasmeen
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Strongly Convex ◽  
Error Estimations ◽  
Integral Identities

In this article, we prove some fractional versions of Hadamard-type inequalities for strongly exponentially α , h − m -convex functions via generalized Riemann–Liouville fractional integrals. The outcomes of this paper provide inequalities of strongly convex, strongly m -convex, strongly s -convex, strongly α , m -convex, strongly s , m -convex, strongly h − m -convex, strongly α , h − m -convex, strongly exponentially convex, strongly exponentially m -convex, strongly exponentially s -convex, strongly exponentially s , m -convex, strongly exponentially h − m -convex, and exponentially α , h − m -convex functions. The error estimations are also studied by applying two fractional integral identities.

scholarly journals Some Fractional Hermite–Hadamard Type Inequalities for Interval-Valued Functions

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 534
Author(s):  
Fangfang Shi ◽  
Guoju Ye ◽  
Dafang Zhao ◽  
Wei Liu
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Liouville Type ◽  
Special Cases ◽  
The Relationship ◽  
Interval Valued

In this paper, firstly we prove the relationship between interval h-convex functions and interval harmonically h-convex functions. Secondly, several new Hermite–Hadamard type inequalities for interval h-convex functions via interval Riemann–Liouville type fractional integrals are established. Finally, we obtain some new fractional Hadamard–Hermite type inequalities for interval harmonically h-convex functions by using the above relationship. Also we discuss the importance of our results and some special cases. Our results extend and improve some previously known results.

scholarly journals Inequalities for generalized Riemann–Liouville fractional integrals of generalized strongly convex functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ghulam Farid ◽  
Young Chel Kwun ◽  
Hafsa Yasmeen ◽  
Abdullah Akkurt ◽  
Shin Min Kang
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Constant Multiplier ◽  
Error Estimations ◽  
Integral Identities

AbstractSome new integral inequalities for strongly $(\alpha ,h-m)$ ( α , h − m ) -convex functions via generalized Riemann–Liouville fractional integrals are established. The outcomes of this paper provide refinements of some fractional integral inequalities for strongly convex, strongly m-convex, strongly $(\alpha ,m)$ ( α , m ) -convex, and strongly $(h-m)$ ( h − m ) -convex functions. Also, the refinements of error estimations of these inequalities are obtained by using two fractional integral identities. Moreover, using a parameter substitution and a constant multiplier, k-fractional versions of established inequalities are also given.

scholarly journals Trapezium-Type Inequalities for Raina’s Fractional Integrals Operator Using Generalized Convex Functions

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1034 ◽  
Author(s):  
Miguel Vivas-Cortez ◽  
Artion Kashuri ◽  
Jorge E. Hernández Hernández
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Fractional Integrals ◽  
Generalized Convex Functions ◽  
Hadamard Inequality ◽  
Trapezium Type ◽  
Special Cases ◽  
The Right ◽  
The Relationship

The authors have reviewed a wide production of scientific articles dealing with the evolution of the concept of convexity and its various applications, and based on this they have detected the relationship that can be established between trapezoidal inequalities, generalized convex functions, and special functions, in particular with the so-called Raina function, which generalizes other better known ones such as the hypergeometric function and the Mittag–Leffler function. The authors approach this situation by studying the Hermite–Hadamard inequality, establishing a useful identity using Raina’s fractional integral operator in the setting of ϕ -convex functions, obtaining some integral inequalities connected with the right-hand side of Hermite–Hadamard-type inequalities for Raina’s fractional integrals. Various special cases have been identified.

scholarly journals New inequalities for F-convex functions pertaining generalized fractional integrals

2020 ◽  
Vol 24 (2) ◽  
pp. 117-131
Author(s):  
Hüseyın Budak ◽  
Pınar Kösem ◽  
Artion Kashuri
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
New Inequalities

In this paper, the authors, utilizing F-convex functions which are defined by B. Samet, establish some new Hermite-Hadamard type inequalities via generalized fractional integrals. Some special cases of our main results recaptured the well-known earlier works.

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