scholarly journals Two point trapezoidal like inequalities involving hypergeometric functions

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2281-2292 ◽  
Author(s):  
Muhammad Awan ◽  
Muhammad Noor ◽  
Marcela Mihai ◽  
Khalida Noor
Keyword(s):  
Differentiable Function ◽  
Convex Functions ◽  
Integral Identity ◽  
Hypergeometric Functions ◽  
Auxiliary Result ◽  
Special Cases

In this paper, we derive a new integral identity for differentiable function. Using this new integral identity as an auxiliary result, we derive some new two point trapezoidal like inequalities for differentiable harmonic h-convex functions. These inequalities can also be viewed as Hermite-Hadamard type inequalities. We also discuss some new special cases which can be deduced from our main results.

scholarly journals A Study of Uniform Harmonic χ -Convex Functions with respect to Hermite-Hadamard’s Inequality and Its Caputo-Fabrizio Fractional Analogue and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Miguel Vivas-Cortez ◽  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Hadamard’S Inequality ◽  
Special Cases ◽  
Harmonic Convex

In this paper, we introduce the notion of uniform harmonic χ -convex functions. We show that this class relates several other unrelated classes of uniform harmonic convex functions. We derive a new version of Hermite-Hadamard’s inequality and its fractional analogue. We also derive a new fractional integral identity using Caputo-Fabrizio fractional integrals. Utilizing this integral identity as an auxiliary result, we obtain new fractional Dragomir-Agarwal type of inequalities involving differentiable uniform harmonic χ -convex functions. We discuss numerous new special cases which show that our results are quite unifying. Finally, in order to show the significance of the main results, we discuss some applications to means of positive real numbers.

scholarly journals A new generalization of some quantum integral inequalities for quantum differentiable convex functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yi-Xia Li ◽  
Muhammad Aamir Ali ◽  
Hüseyin Budak ◽  
Mujahid Abbas ◽  
Yu-Ming Chu
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
Quantum Integral ◽  
Special Cases

AbstractIn this paper, we offer a new quantum integral identity, the result is then used to obtain some new estimates of Hermite–Hadamard inequalities for quantum integrals. The results presented in this paper are generalizations of the comparable results in the literature on Hermite–Hadamard inequalities. Several inequalities, such as the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint–trapezoid-like integral inequality, and the trapezoid-like integral inequality, are obtained as special cases of our main results.

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 Simpson type inequalities and applications

2021 ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Michael Th. Rassias ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Quadrature Formula ◽  
Convex Functions ◽  
Integral Identity ◽  
Higher Order ◽  
Auxiliary Result ◽  
First Order ◽  
New Class ◽  
Differentiable Functions

AbstractA new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of $$\sigma >0$$ σ > 0 . We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.

scholarly journals Some New Postquantum Integral Inequalities

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yu-Ming Chu ◽  
Muhammad Uzair Awan ◽  
Sadia Talib ◽  
Sabah Iftikhar ◽  
Latifa Riahi
Keyword(s):  
Integral Identity ◽  
Integral Inequalities ◽  
Auxiliary Result ◽  
Special Cases ◽  
Preinvex Functions

The goal of this paper is to derive a new generalized postquantum integral identity. Using this new identity as an auxiliary result, we derive some new variants of integral inequalities using p , q -differentiable preinvex functions. We also point out some special cases of the obtained results which show that our results are quite unifying ones.

scholarly journals New Inequalities for Strongly r-Convex Functions

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Huriye Kadakal
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
Special Cases ◽  
Class Of Functions ◽  
New Inequalities

In this study, firstly we introduce a new concept called “strongly r-convex function.” After that we establish Hermite-Hadamard-like inequalities for this class of functions. Moreover, by using an integral identity together with some well known integral inequalities, we establish several new inequalities for n-times differentiable strongly r-convex functions. In special cases, the results obtained coincide with the well-known results in the literature.

scholarly journals Some new Caputo fractional derivative inequalities for exponentially $ (\theta, h-m) $–convex functions

2021 ◽  
Vol 7 (2) ◽  
pp. 3006-3026
Author(s):  
Imran Abbas Baloch ◽  
◽  
Thabet Abdeljawad ◽  
Sidra Bibi ◽  
Aiman Mukheimer ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Convex Functions ◽  
Integral Identity ◽  
Fractional Derivatives ◽  
Order Derivative ◽  
Special Cases

<abstract><p>Firstly, we obtain some inequalities of Hadamard type for exponentially $ (\theta, h-m) $–convex functions via Caputo $ k $–fractional derivatives. Secondly, using integral identity including the $ (n+1) $–order derivative of a given function via Caputo $ k $-fractional derivatives we prove some of its related results. Some new results are given and known results are recaptured as special cases from our results.</p></abstract>

scholarly journals Some new inequalities for generalized convex functions pertaining generalized fractional integral operators and their applications

2021 ◽  
Vol 17 (1) ◽  
pp. 37-64
Author(s):  
A. Kashuri ◽  
M.A. Ali ◽  
M. Abbas ◽  
M. Toseef
Keyword(s):  
Differentiable Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Generalized Convex Functions ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators ◽  
New Inequalities

Abstract In this paper, authors establish a new identity for a differentiable function using generic integral operators. By applying it, some new integral inequalities of trapezium, Ostrowski and Simpson type are obtained. Moreover, several special cases have been studied in detail. Finally, many useful applications have been found.

scholarly journals On new generalized unified bounds via generalized exponentially harmonically s-convex functions on fractal sets

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yu-Ming Chu ◽  
Saima Rashid ◽  
Thabet Abdeljawad ◽  
Aasma Khalid ◽  
Humaira Kalsoom
Keyword(s):  
Convex Functions ◽  
Fractal Theory ◽  
Type Inequality ◽  
Auxiliary Result ◽  
Fractal Sets ◽  
Special Cases ◽  
Mathematical Sciences ◽  
Novel Strategy ◽  
Real Linear ◽  
Novel Applications

AbstractThe visual beauty reflects the practicability and superiority of design dependent on the fractal theory. Based on the applicability in practice, it shows that it is the completely feasible, self-comparability and multifaceted nature of fractal sets that made it an appealing field of research. There is a strong correlation between fractal sets and convexity due to its intriguing nature in the mathematical sciences. This paper investigates the notions of generalized exponentially harmonically ($GEH$ G E H ) convex and $GEH$ G E H s-convex functions on a real linear fractal sets $\mathbb{R}^{\alpha }$ R α ($0<\alpha \leq 1$ 0 < α ≤ 1 ). Based on these novel ideas, we derive the generalized Hermite–Hadamard inequality, generalized Fejér–Hermite–Hadamard type inequality and Pachpatte type inequalities for $GEH$ G E H s-convex functions. Taking into account the local fractal identity; we establish a certain generalized Hermite–Hadamard type inequalities for local differentiable $GEH$ G E H s-convex functions. Meanwhile, another auxiliary result is employed to obtain the generalized Ostrowski type inequalities for the proposed techniques. Several special cases of the proposed concept are presented in the light of generalized exponentially harmonically convex, generalized harmonically convex and generalized harmonically s-convex. Meanwhile, an illustrative example and some novel applications in generalized special means are obtained to ensure the correctness of the present results. This novel strategy captures several existing results in the corresponding literature. Finally, we suppose that the consequences of this paper can stimulate those who are interested in fractal analysis.

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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