scholarly journals Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1666
Author(s):  
Surang Sitho ◽  
Muhammad Aamir Ali ◽  
Hüseyin Budak ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Type Inequality ◽  
Quantum Calculus ◽  
Proper Form ◽  
Quantum Integral ◽  
Preinvex Functions

In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the qϰ2-quantum integral to show midpoint and trapezoidal inequalities for qϰ2-differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via qϰ1-quantum integral is not valid for preinvex functions, and we present its proper form. We use qϰ1-quantum integrals to show midpoint inequalities for qϰ1-differentiable preinvex functions. It is also demonstrated that by considering the limit q→1− and ηϰ2,ϰ1=−ηϰ1,ϰ2=ϰ2−ϰ1 in the newly derived results, the newly proved findings can be turned into certain known results.

scholarly journals Quantum Integral Inequalities with Respect to Raina’s Function via Coordinated Generalized Ψ -Convex Functions with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Saima Rashid ◽  
Saad Ihsan Butt ◽  
Shazia Kanwal ◽  
Hijaz Ahmad ◽  
Miao-Kun Wang
Keyword(s):  
Quantum Theory ◽  
Real World ◽  
Convex Functions ◽  
Integral Identity ◽  
Special Functions ◽  
Type Inequality ◽  
The Novel ◽  
Quantum Calculus ◽  
Quantum Integral ◽  
Formula Type

In accordance with the quantum calculus, we introduced the two variable forms of Hermite-Hadamard- ( H H -) type inequality over finite rectangles for generalized Ψ -convex functions. This novel framework is the convolution of quantum calculus, convexity, and special functions. Taking into account the q ^ 1 q ^ 2 -integral identity, we demonstrate the novel generalizations of the H H -type inequality for q ^ 1 q ^ 2 -differentiable function by acquainting Raina’s functions. Additionally, we present a different approach that can be used to characterize H H -type variants with respect to Raina’s function of coordinated generalized Ψ -convex functions within the quantum techniques. This new study has the ability to generate certain novel bounds and some well-known consequences in the relative literature. As application viewpoint, the proposed study for changing parametric values associated with Raina’s functions exhibits interesting results in order to show the applicability and supremacy of the obtained results. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.

New generalized 2D Ostrowski type inequalities on time scales with k2 points using a parameter

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3155-3169 ◽  
Author(s):  
Seth Kermausuor ◽  
Eze Nwaeze
Keyword(s):  
Numerical Analysis ◽  
Time Scales ◽  
Type Inequality ◽  
Dimensional Case ◽  
Multiple Points ◽  
Quantum Calculus ◽  
Independent Variables ◽  
Ostrowski Type Inequality ◽  
Two Independent Variables

Recently, a new Ostrowski type inequality on time scales for k points was proved in [G. Xu, Z. B. Fang: A Generalization of Ostrowski type inequality on time scales with k points. Journal of Mathematical Inequalities (2017), 11(1):41-48]. In this article, we extend this result to the 2-dimensional case. Besides extension, our results also generalize the three main results of Meng and Feng in the paper [Generalized Ostrowski type inequalities for multiple points on time scales involving functions of two independent variables. Journal of Inequalities and Applications (2012), 2012:74]. In addition, we apply some of our theorems to the continuous, discrete, and quantum calculus to obtain more interesting results in this direction. We hope that results obtained in this paper would find their place in approximation and numerical analysis.

scholarly journals New Integral Inequalities via Generalized Preinvex Functions

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 296
Author(s):  
Muhammad Tariq ◽  
Asif Ali Shaikh ◽  
Soubhagya Kumar Sahoo ◽  
Hijaz Ahmad ◽  
Thanin Sitthiwirattham ◽  
...  
Keyword(s):  
Integral Inequality ◽  
Type Inequality ◽  
Integral Inequalities ◽  
Power Mean ◽  
Science And Engineering ◽  
Special Means ◽  
Algebraic Properties ◽  
Convexity Theory ◽  
Hölder’S Integral Inequality ◽  
Preinvex Functions

The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m–preinvex function and to present inequalities of the Hermite–Hadamard type in the setting of the generalized s-type m–preinvex function. First, we give the definition and then investigate some of its algebraic properties and examples. We also present some refinements of the Hermite–Hadamard-type inequality using Hölder’s integral inequality, the improved power-mean integral inequality, and the Hölder-İşcan integral inequality. Finally, some results for special means are deduced. The results established in this paper can be considered as the generalization of many published results of inequalities and convexity theory.

scholarly journals On Weighted Montgomery Identity for k Points and Its Associates on Time Scales

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Eze R. Nwaeze ◽  
Ana M. Tameru
Keyword(s):  
Weight Function ◽  
Time Scales ◽  
Type Inequality ◽  
Montgomery Identity ◽  
Quantum Calculus ◽  
Ostrowski Type Inequality

The purpose of this paper is to establish a weighted Montgomery identity for k points and then use this identity to prove a new weighted Ostrowski type inequality. Our results boil down to the results of Liu and Ngô if we take the weight function to be the identity map. In addition, we also generalize an inequality of Ostrowski-Grüss type on time scales for k points. For k=2, we recapture a result of Tuna and Daghan. Finally, we apply our results to the continuous, discrete, and quantum calculus to obtain more results in this direction.

scholarly journals Some New q—Integral Inequalities Using Generalized Quantum Montgomery Identity via Preinvex Functions

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 553 ◽  
Author(s):  
Miguel Vivas-Cortez ◽  
Artion Kashuri ◽  
Rozana Liko ◽  
Jorge E. Hernández Hernández
Keyword(s):  
Integral Inequalities ◽  
Montgomery Identity ◽  
Quantum Calculus ◽  
Special Cases ◽  
Preinvex Functions

In this work the authors establish a new generalized version of Montgomery’s identity in the setting of quantum calculus. From this result, some new estimates of Ostrowski type inequalities are given using preinvex functions. Given the generality of preinvex functions, particular q —integral inequalities are established with appropriate choice of the parametric bifunction. Some new special cases from the main results are obtained and some known results are recaptured as well. At the end, a briefly conclusion is given.

scholarly journals On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Yu-Ming Chu
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Shift Operator ◽  
Type Inequality ◽  
Fractional Quantum ◽  
Quantum Calculus ◽  
New Directions

In the article, we present several generalizations for the generalized Čebyšev type inequality in the frame of quantum fractional Hahn’s integral operator by using the quantum shift operator  σΨqς=qς+1−qσς∈l1,l2,σ=l1+ω/1−q,0<q<1,ω≥0. As applications, we provide some associated variants to illustrate the efficiency of quantum Hahn’s integral operator and compare our obtained results and proposed technique with the previously known results and existing technique. Our ideas and approaches may lead to new directions in fractional quantum calculus theory.

Some quantum integral inequalities via preinvex functions

2015 ◽  
Vol 269 ◽  
pp. 242-251 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Muhammad Uzair Awan
Keyword(s):  
Integral Inequalities ◽  
Quantum Integral ◽  
Preinvex Functions

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 Certain Generalized Quantum Simpson's and Quantum Newton's type Inequalities for Convex Functions in Quantum Calculus

2020 ◽  
Author(s):  
Muhammad Aamir Ali ◽  
Hüseyin BUDAK ◽  
PRAVEEN AGARWAL ◽  
Yuming Chu
Keyword(s):  
Convex Functions ◽  
Integral Inequalities ◽  
Quantum Calculus ◽  
Quantum Integral ◽  
Classical Integral

In this paper first we present some new identities by using the notions of quantum integrals and derivatives which allows us to obtain new quantum Simpson’s and quantum Newton’s type inequalities for differentiable convex functions by using the q_{x}-quantum integral and q^{y}-quantum integral. In particular, this paper generalises and extends previous results obtained by the various authors in the field of quantum and classical integral inequalities.

scholarly journals Post Quantum Integral Inequalities of Hermite-Hadamard-Type Associated with Co-Ordinated Higher-Order Generalized Strongly Pre-Invex and Quasi-Pre-Invex Mappings

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 443 ◽  
Author(s):  
Humaira Kalsoom ◽  
Saima Rashid ◽  
Muhammad Idrees ◽  
Farhat Safdar ◽  
Saima Akram ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Research Study ◽  
Type Inequality ◽  
Higher Order ◽  
Auxiliary Result ◽  
Contemporary Theory ◽  
Invex Set ◽  
Quantum Integral ◽  
Novel Variants ◽  
Explicit Bounds

By using the contemporary theory of inequalities, this study is devoted to proposing a number of refinements inequalities for the Hermite-Hadamard’s type inequality and conclude explicit bounds for two new definitions of ( p 1 p 2 , q 1 q 2 ) -differentiable function and ( p 1 p 2 , q 1 q 2 ) -integral for two variables mappings over finite rectangles by using pre-invex set. We have derived a new auxiliary result for ( p 1 p 2 , q 1 q 2 ) -integral. Meanwhile, by using the symmetry of an auxiliary result, it is shown that novel variants of the the Hermite-Hadamard type for ( p 1 p 2 , q 1 q 2 ) -differentiable utilizing new definitions of generalized higher-order strongly pre-invex and quasi-pre-invex mappings. It is to be acknowledged that this research study would develop new possibilities in pre-invex theory, quantum mechanics and special relativity frameworks of varying nature for thorough investigation.

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