Vectorial fractional integral inequalities with convexity

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
George Anastassiou
Keyword(s):  
Integral Inequalities ◽  
Fractional Integrals ◽  
General Integral ◽  
Hardy Type ◽  
Wide Range ◽  
Left And Right ◽  
Multiple Integrals ◽  
P Type ◽  
Derivatives Of ◽  
Increasing Functions

AbstractHere we present vectorial general integral inequalities involving products of multivariate convex and increasing functions applied to vectors of functions. As specific applications we derive a wide range of vectorial fractional inequalities of Hardy type. These involve the left and right: Erdélyi-Kober fractional integrals, mixed Riemann-Liouville fractional multiple integrals. Next we produce multivariate Poincaré type vectorial fractional inequalities involving left fractional radial derivatives of Canavati type, Riemann-Liouville and Caputo types. The exposed inequalities are of L p type, p ≥ 1, and exponential type.

Some new generalizations for GA-convex functions

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
Ahmet Akdemir ◽  
Özdemir Emin ◽  
Ardıç Avcı ◽  
Abdullatif Yalçın
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
General Integral ◽  
Second Derivatives ◽  
Derivatives Of

In this paper, firstly we prove an integral identity that one can derive several new equalities for special selections of n from this identity: Secondly, we established more general integral inequalities for functions whose second derivatives of absolute values are GA-convex functions based on this equality.

scholarly journals Riemann-Liouville Fractional integral operators with respect to increasing functions and strongly $ (\alpha, m) $-convex functions

2021 ◽  
Vol 6 (10) ◽  
pp. 11403-11424
Author(s):  
Ghulam Farid ◽  
◽  
Hafsa Yasmeen ◽  
Hijaz Ahmad ◽  
Chahn Yong Jung ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Fractional Integral Operators ◽  
Strongly Convex ◽  
Increasing Functions ◽  
Integral Identities

<abstract><p>In this paper Hadamard type inequalities for strongly $ (\alpha, m) $-convex functions via generalized Riemann-Liouville fractional integrals are studied. These inequalities provide generalizations as well as refinements of several well known inequalities. The established results are further connected with fractional integral inequalities for Riemann-Liouville fractional integrals of convex, strongly convex and strongly $ m $-convex functions. By using two fractional integral identities some more Hadamard type inequalities are proved.</p></abstract>

scholarly journals Estimation of Integral Inequalities Using the Generalized Fractional Derivative Operator in the Hilfer Sense

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Saima Rashid ◽  
Rehana Ashraf ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Evaluation Process ◽  
Strong Relationship ◽  
Integral Inequalities ◽  
Practical Significance ◽  
Caputo Fractional Derivatives ◽  
Inequality Theory ◽  
Left And Right ◽  
Derivatives Of

With the great progress of fractional calculus, integral inequalities have been greatly enriched by fractional operators; users and researchers have formed a real-world phenomenon in the production of the evaluation process, which results in convexity. Monotonicity and inequality theory has a strong relationship, whichever we work on, and we can apply it to the other one due to the strong correlation produced between them, especially in the past few years. In this article, we introduce some estimations of left and right sides of the generalized Caputo fractional derivatives of a function for n th order differentiability via convex function, and related inequalities have been presented. Monotonicity and convexity of functions are used with some usual and straightforward inequalities. Moreover, we establish some new inequalities for C ⌣ eby s ⌣ ev and Gr u ¨ ss type involving the generalized Caputo fractional derivative operators. The finding provides the theoretical basis and practical significance for the establishment of fractional calculus in convexity. It also introduces new ways of thinking and methods for innovative scientific research.

scholarly journals Fractional integral inequalities of Hermite-Hadamard type for convex functions with respect to a monotone function

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2401-2411
Author(s):  
Pshtiwan Mohammed
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Monotone Function ◽  
Type Inequality ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Increasing Functions

In the literature, the left-side of Hermite-Hadamard?s inequality is called a midpoint type inequality. In this article, we obtain new integral inequalities of midpoint type for Riemann-Liouville fractional integrals of convex functions with respect to increasing functions. The resulting inequalities generalize some recent integral inequalities and Riemann-Liouville fractional integral inequalities established in earlier works. Finally, applications of our work are demonstrated via the known special functions.

scholarly journals GENERALIZED MULTIVARIATE PRABHAKAR TYPE FRACTIONAL INTEGRALS AND INEQUALITIES

2021 ◽  
Vol 12 (4) ◽  
pp. 1-15
Author(s):  
GEORGE A. ANASTASSIOU
Keyword(s):  
Linear Operators ◽  
Positive Linear Operators ◽  
Fractional Integrals ◽  
Hardy Type Inequalities ◽  
Basic Properties ◽  
Hardy Type ◽  
Left And Right

We introduce here the mixed generalized multivariate Prabhakar type left and right fractional integrals and study their basic properties, such as preservation of continuity and their boundedness as positive linear operators. Then we produce an interesting variety of related multivariate left and right fractional Hardy type inequalities under convexity. We introduce also other related multivariate fractional integrals

scholarly journals New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
İmdat İşcan
Keyword(s):  
Integral Inequalities ◽  
Fractional Integrals ◽  
Real Numbers ◽  
Positive Real ◽  
General Integral ◽  
Lipschitzian Functions ◽  
Special Means ◽  
Hadamard Fractional Integrals

The author obtains new estimates on generalization of Hadamard, Ostrowski, and Simpson type inequalities for Lipschitzian functions via Hadamard fractional integrals. Some applications to special means of positive real numbers are also given.

scholarly journals Some generalization of integral inequalities for twice differentiable mappings involving fractional integrals

2015 ◽  
Vol 7 (2) ◽  
pp. 251-264
Author(s):  
Mehmet Zeki Sarikaya ◽  
Huseyin Budak
Keyword(s):  
Integral Identity ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
General Integral ◽  
Differentiable Mappings

Abstract In this paper, a general integral identity involving Riemann- Liouville fractional integrals is derived. By use this identity, we establish new some generalized inequalities of the Hermite-Hadamard’s type for functions whose absolute values of derivatives are convex.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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