scholarly journals q -Hermite–Hadamard Inequalities for Generalized Exponentially s , m ; η -Preinvex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Hua Wang ◽  
Humaira Kalsoom ◽  
Hüseyin Budak ◽  
Muhammad Idrees
Keyword(s):  
Differentiable Function ◽  
Integral Identity ◽  
Special Cases ◽  
Preinvex Functions ◽  
Definition Of ◽  
Classical Convexity

In this article, we introduce a new extension of classical convexity which is called generalized exponentially s , m ; η -preinvex functions. Also, it is seen that the new definition of generalized exponentially s , m ; η -preinvex functions describes different new classes as special cases. To prove our main results, we derive a new q m κ 2 -integral identity for the twice q m κ 2 -differentiable function. By using this identity, we show essential new results for Hermite–Hadamard-type inequalities for the q m κ 2 -integral by utilizing differentiable exponentially s , m ; η -preinvex functions. The results presented in this article are unification and generalization of the comparable results in the literature.

scholarly journals Two-Variable Quantum Integral Inequalities of Simpson-Type Based on Higher-Order Generalized Strongly Preinvex and Quasi-Preinvex Functions

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 51 ◽  
Author(s):  
Humaira Kalsoom ◽  
Saima Rashid ◽  
Muhammad Idrees ◽  
Yu-Ming Chu ◽  
Dumitru Baleanu
Keyword(s):  
Integral Identity ◽  
Higher Order ◽  
Integral Inequalities ◽  
Numerical Approximations ◽  
New Class ◽  
Quantum Integral ◽  
Special Cases ◽  
Preinvex Functions ◽  
Definition Of

In this paper, we present a new definition of higher-order generalized strongly preinvex functions. Moreover, it is observed that the new class of higher-order generalized strongly preinvex functions characterize various new classes as special cases. We acquire a new q 1 q 2 -integral identity, then employing this identity, we establish several two-variable q 1 q 2 -integral inequalities of Simpson-type within a class of higher-order generalized strongly preinvex and quasi-preinvex functions. Finally, the utilities of our numerical approximations have concrete applications.

scholarly journals A new q-integral identity and estimation of its bounds involving generalized exponentially μ-preinvex functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Sadia Talib ◽  
Artion Kashuri ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
...  
Keyword(s):  
Differentiable Function ◽  
Integral Identity ◽  
Second Order ◽  
Integral Inequalities ◽  
Absolute Value ◽  
Type Integral ◽  
Preinvex Functions

Abstract In the article, we introduce the generalized exponentially μ-preinvex function, derive a new q-integral identity for second order q-differentiable function, and establish several new q-trapezoidal type integral inequalities for the function whose absolute value of second q-derivative is exponentially μ-preinvex.

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 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 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 Quantum Integral Inequalities of Simpson-Type for Strongly Preinvex Functions

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 751 ◽  
Author(s):  
Yongping Deng ◽  
Muhammad Uzair Awan ◽  
Shanhe Wu
Keyword(s):  
Integral Identity ◽  
Integral Inequalities ◽  
Quantum Integral ◽  
Special Cases ◽  
Preinvex Functions

In this paper, we establish a new q-integral identity, the result is then used to derive two q-integral inequalities of Simpson-type involving strongly preinvex functions. Some special cases of the obtained results are also considered, it is shown that several new and previously known results can be derived via generalized strongly preinvex functions and quantum integrals.

scholarly journals New Hermite-Hadamard Type Inequalities forn-Times Differentiable ands-Logarithmically Preinvex Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Shuhong Wang ◽  
Ximin Liu
Keyword(s):  
Differentiable Function ◽  
Integral Identity ◽  
Preinvex Functions

The concept ofs-logarithmically preinvex function is introduced, and by creating an integral identity involving ann-times differentiable function, some new Hermite-Hadamard type inequalities fors-logarithmically preinvex functions are established.

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 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 Final coalgebras as greatest fixed points in ZF set theory

1999 ◽  
Vol 9 (5) ◽  
pp. 545-567 ◽  
Author(s):  
LAWRENCE C. PAULSON
Keyword(s):  
Fixed Points ◽  
Set Theory ◽  
Theorem Prover ◽  
Infinite Length ◽  
Special Cases ◽  
Definition Of ◽  
Recursive Definitions ◽  
Ordered Pairs

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo–Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's solution and substitution lemmas are proved in the style of Rutten and Turi (1993). The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint.Compared with previous work (Paulson, 1995a), iterated substitutions and solutions are considered, as well as final coalgebras defined with respect to parameters. The disjoint sum construction is replaced by a smoother treatment of urelements that simplifies many of the derivations.The theory facilitates machine implementation of recursive definitions by letting both inductive and coinductive definitions be represented as fixed points. It has already been applied to the theorem prover Isabelle (Paulson, 1994).

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