scholarly journals Fractional Integral Inequalities for Strongly h -Preinvex Functions for a kth Order Differentiable Functions

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1448 ◽  
Author(s):  
Saima Rashid ◽  
Muhammad Amer Latif ◽  
Zakia Hammouch ◽  
Yu-Ming Chu
Keyword(s):  
Integral Operator ◽  
Differentiable Function ◽  
Real World ◽  
Fractional Integral ◽  
Higher Order ◽  
Fractional Integrals ◽  
Differentiable Functions ◽  
Real World Applications ◽  
Mathematical Problems ◽  
Preinvex Functions

The objective of this paper is to derive Hermite-Hadamard type inequalities for several higher order strongly h -preinvex functions via Riemann-Liouville fractional integrals. These results are the generalizations of the several known classes of preinvex functions. An identity associated with k-times differentiable function has been established involving Riemann-Liouville fractional integral operator. A number of new results can be deduced as consequences for the suitable choices of the parameters h and σ . Our outcomes with these new generalizations have the abilities to be implemented for the evaluation of many mathematical problems related to real world applications.

scholarly journals Quantum Hermite-Hadamard type inequalities for generalized strongly preinvex functions

2021 ◽  
Vol 6 (12) ◽  
pp. 13291-13310
Author(s):  
Humaira Kalsoom ◽  
◽  
Muhammad Amer Latif ◽  
Muhammad Idrees ◽  
Muhammad Arif ◽  
...  
Keyword(s):  
Real World ◽  
Higher Order ◽  
Integral Type ◽  
The Real ◽  
Quantum Calculus ◽  
Special Cases ◽  
Mathematical Problems ◽  
Preinvex Functions ◽  
Function Approximations ◽  
Integral Identities

<abstract><p>In accordance with the quantum calculus, the quantum Hermite-Hadamard type inequalities shown in recent findings provide improvements to quantum Hermite-Hadamard type inequalities. We acquire a new $ q{_{\kappa_1}} $-integral and $ q{^{\kappa_2}} $-integral identities, then employing these identities, we establish new quantum Hermite-Hadamard $ q{_{\kappa_1}} $-integral and $ q{^{\kappa_2}} $-integral type inequalities through generalized higher-order strongly preinvex and quasi-preinvex functions. The claim of our study has been graphically supported, and some special cases are provided as well. Finally, we present a comprehensive application of the newly obtained key results. Our outcomes from these new generalizations can be applied to evaluate several mathematical problems relating to applications in the real world. These new results are significant for improving integrated symmetrical function approximations or functions of some symmetry degree.</p></abstract>

scholarly journals Estimations of Upper Bounds for n -th Order Differentiable Functions Involving χ -Riemann–Liouville Integrals via γ -Preinvex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Sadia Talib ◽  
Muhammad Uzair Awan
Keyword(s):  
Integral Identity ◽  
Fractional Integral ◽  
Upper Bounds ◽  
Fractional Integrals ◽  
Differentiable Functions ◽  
Special Cases ◽  
Preinvex Functions

A new fractional integral identity is obtained involving n -th order differentiable functions and χ -Riemann–Liouville fractional integrals. Then, some associated estimates of upper bounds involving γ -preinvex functions are obtained. In order to relate some unrelated results, several special cases are discussed.

scholarly journals Some Ostrowski type inequalities for mappings whose second derivatives are preinvex function via fractional integral operator

2021 ◽  
Vol 7 (3) ◽  
pp. 3303-3320
Author(s):  
Jamshed Nasir ◽  
◽  
Shahid Qaisar ◽  
Saad Ihsan Butt ◽  
Ather Qayyum ◽  
...  
Keyword(s):  
Applied Sciences ◽  
Integral Operator ◽  
Fractional Integral ◽  
Bessel Functions ◽  
Fractional Integral Operator ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Differentiable Functions ◽  
Second Derivatives ◽  
Preinvex Functions

<abstract><p>The comprehension of inequalities in preinvexity is very important for studying fractional calculus and its effectiveness in many applied sciences. In this article, we develop and study of fractional integral inequalities whose second derivatives are preinvex functions. We investigate and prove new lemma for twice differentiable functions involving Riemann-Liouville(R-L) fractional integral operator. On the basis of this newly developed lemma, we make some new results regarding of this identity. These new results yield us some generalizations of the prior results. This study builds upon on a novel new auxiliary result which enables us to develop new variants of Ostrowski type inequalities for twice differentiable preinvex mappings. As an application, several estimates concerning Bessel functions of real numbers are also illustrated.</p></abstract>

scholarly journals Fractional trapezium type inequalities for twice differentiable preinvex functions and their applications

2020 ◽  
Vol 10 (2) ◽  
pp. 226-236
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Integral Operator ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Recent Past ◽  
Trapezium Type ◽  
Special Means ◽  
Special Cases ◽  
Type Integral ◽  
Preinvex Functions ◽  
Almost All

Trapezoidal inequalities for functions of divers natures are useful in numerical computations. The authors have proved an identity for a generalized integral operator via twice differentiable preinvex function. By applying the established identity, the generalized trapezoidal type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of presented results to special means have been analyzed. The ideas and techniques of this paper may stimulate further research.

scholarly journals New Applications of the Bernardi Integral Operator

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1180
Author(s):  
Shigeyoshi Owa ◽  
H. Özlem Güney
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Closed Unit Disk ◽  
New Applications

Let A ( p , n ) be the class of f ( z ) which are analytic p-valent functions in the closed unit disk U ¯ = z ∈ C : z ≤ 1 . The expression B − m − λ f ( z ) is defined by using fractional integrals of order λ for f ( z ) ∈ A ( p , n ) . When m = 1 and λ = 0 , B − 1 f ( z ) becomes Bernardi integral operator. Using the fractional integral B − m − λ f ( z ) , the subclass T p , n α s , β , ρ ; m , λ of A ( p , n ) is introduced. In the present paper, we discuss some interesting properties for f ( z ) concerning with the class T p , n α s , β , ρ ; m , λ . Also, some interesting examples for our results will be considered.

scholarly journals The higher order commutators of the fractional integrals on Hardy spaces

2005 ◽  
Vol 79 (1) ◽  
pp. 11-24
Author(s):  
Shunchao Long ◽  
Jian Wang
Keyword(s):  
Hardy Spaces ◽  
Fractional Integral ◽  
Higher Order ◽  
Type Operator ◽  
Fractional Integrals ◽  
Integral Type ◽  
Bmo Function

AbstractIn this paper we investigate the boundedness on Hardy spaces for the higher order commutator Tb, m generated by the BMO function b and fractional integral type operator Tτ, and establish the boundness theorems for Tτb, m from Hp1.q1.sb, m to Lp2 and to Hp2 (0 < p1 ≤ 1), and from H Ka. p1.sq1, b, m to Ka.p2q2 and to H Ka. p2q2, respectively, for certain ranges of α, p1, q1, p2, q2 and s.

scholarly journals A basic study of a fractional integral operator with extended Mittag-Leffler kernel

2021 ◽  
Vol 6 (11) ◽  
pp. 12757-12770
Author(s):  
Gauhar Rahman ◽  
◽  
Iyad Suwan ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad ◽  
...  
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Fractional Integrals ◽  
Basic Study ◽  
Basic Properties

<abstract><p>In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.</p></abstract>

scholarly journals Bounds for the Remainder in Simpson’s Inequality via n-Polynomial Convex Functions of Higher Order Using Katugampola Fractional Integrals

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yu-Ming Chu ◽  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javad ◽  
Awais Gul Khan
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Fractional Integral ◽  
Higher Order ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Simpson’S Inequality ◽  
Generalized Fractional Integral

The goal of this paper is to derive some new variants of Simpson’s inequality using the class of n-polynomial convex functions of higher order. To obtain the main results of the paper, we first derive a new generalized fractional integral identity utilizing the concepts of Katugampola fractional integrals. This new fractional integral identity will serve as an auxiliary result in the development of the main results of this paper.

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