scholarly journals New Multi-Parametrized Estimates Having pth-Order Differentiability in Fractional Calculus for Predominating ℏ-Convex Functions in Hilbert Space

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 222 ◽  
Author(s):  
Saima Rashid ◽  
Humaira Kalsoom ◽  
Zakia Hammouch ◽  
Rehana Ashraf ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Convex Function ◽  
Convex Functions ◽  
Higher Order ◽  
Quasiconvex Functions ◽  
Auxiliary Result ◽  
Data Segmentation ◽  
Special Cases ◽  
Novel Variants ◽  
Convexity Theory

In Hilbert space, we develop a novel framework to study for two new classes of convex function depending on arbitrary non-negative function, which is called a predominating ℏ-convex function and predominating quasiconvex function, with respect to η , are presented. To ensure the symmetry of data segmentation and with the discussion of special cases, it is shown that these classes capture other classes of η -convex functions, η -quasiconvex functions, strongly ℏ-convex functions of higher-order and strongly quasiconvex functions of a higher order, etc. Meanwhile, an auxiliary result is proved in the sense of κ -fractional integral operator to generate novel variants related to the Hermite–Hadamard type for p t h -order differentiability. It is hoped that this research study will open new doors for in-depth investigation in convexity theory frameworks of a varying nature.

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 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 Hermite-Hadamard type inequalities for (P;m)-function and quasi m-convex functions

2020 ◽  
Vol 10 (1) ◽  
pp. 78-84
Author(s):  
Mahir Kadakal
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
New Class ◽  
Special Cases ◽  
First Time ◽  
Class Of Functions

In this paper, we introduce a new class of functions called as (P;m)-function and quasi-m-convex function. Some inequalities of Hadamard's type for these functions are given. Some special cases are discussed. Results represent signicant renement and improvement of the previous results. We should especially mention that the denition of (P;m)-function and quasi-m-convexity are given for the first time in the literature and moreover, the results obtained in special cases coincide with thewell-known results in the literature.

scholarly journals Some inequalities for strongly $(p,h)$-harmonic convex functions

2019 ◽  
Vol 11 (1) ◽  
pp. 119-135
Author(s):  
M.A. Noor ◽  
K.I. Noor ◽  
S. Iftikhar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
New Class ◽  
Special Cases ◽  
Harmonic Convex

In this paper, we show that harmonic convex functions $f$ is strongly $(p, h)$-harmonic convex functions if and only if it can be decomposed as $g(x) = f(x) - c (\frac{1}{x^p})^2,$ where $g(x)$ is $(p, h)$-harmonic convex function. We obtain some new estimates  class of strongly $(p, h)$-harmonic convex functions involving hypergeometric and beta functions. As applications of our results, several important special cases are discussed. We also introduce a new class of harmonic convex functions, which is called strongly $(p, h)$-harmonic $\log$-convex functions. Some new Hermite-Hadamard type inequalities for strongly $(p, h)$-harmonic $log$-convex functions are obtained. These results  can be viewed as important refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.

scholarly journals Some inequalities for geometrically-arithmetically h-convex functions

2014 ◽  
Vol 23 (1) ◽  
pp. 91-98
Author(s):  
MUHAMMAD ASLAM NOOR ◽  
◽  
KHALIDA INAYAT NOOR ◽  
MUHAMMAD UZAIR AWAN ◽  
◽  
...  
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Special Cases ◽  
Geometrically Convex Function

In this paper, we consider a class of geometrically convex function which is called geometrically-arithmetically h-convex function. Some inequalities of Hermite-Hadamard type for geometrically-arithmetically h-convex functions are derived. Several special cases are discussed.

scholarly journals Some integral inequalities for approximately h—convex functions and their applications

2021 ◽  
Vol 40 (2) ◽  
pp. 481-504
Author(s):  
Artion Kashuri ◽  
Muhammad Raees ◽  
Matloob Anwar
Keyword(s):  
Convex Function ◽  
Integral Inequality ◽  
Convex Functions ◽  
Integral Inequalities ◽  
Integral Type ◽  
Special Cases ◽  
Hölder’S Integral Inequality ◽  
Error Estimations

In this paper, by applying the new and improved form of Hölder’s integral inequality called Hölder—Íşcan integral inequality three inequalities of Hermite—Hadamard and Hadamard integral type for (h, d)—convex functions have been established. Various special cases including classes for instance, h—convex, s—convex function of Breckner and Godunova—Levin—Dragomir and strong versions of the aforementioned types of convex functions have been identified. Some applications to error estimations of presented results have been analyzed. At the end, a briefly conclusion is given.

scholarly journals Strongly Convex Functions of Higher Order Involving Bifunction

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1028 ◽  
Author(s):  
Bandar B. Mohsen ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Mihai Postolache
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Higher Order ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
New Concepts ◽  
Special Cases ◽  
Novel Applications

Some new concepts of the higher order strongly convex functions involving an arbitrary bifuction are considered in this paper. Some properties of the higher order strongly convex functions are investigated under suitable conditions. Some important special cases are discussed. The parallelogram laws for Banach spaces are obtained as applications of higher order strongly affine convex functions as novel applications. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

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.

Generalization of cyclic refinements of Jensen inequality by montgomery identity and Green’s function

2018 ◽  
Vol 11 (04) ◽  
pp. 1850060 ◽  
Author(s):  
Nasir Mehmood ◽  
Saad Ihsan Butt ◽  
Josip Pečarić
Keyword(s):  
Convex Function ◽  
Green’S Function ◽  
Green's Function ◽  
Convex Functions ◽  
Mean Value ◽  
Higher Order ◽  
Jensen Inequality ◽  
Linear Functionals ◽  
Montgomery Identity ◽  
Exponential Convexity

We consider discrete and continuous cyclic refinements of Jensen’s inequality and generalize them from convex function to higher order convex function by means of Lagrange Green’s function and Montgomery identity. We give application of our results by formulating the monotonicity of the linear functionals obtained from generalized identities utilizing the theory of inequalities for [Formula: see text]-convex functions at a point. We compute Grüss and Ostrowski type bounds for generalized identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity log convexity and mean value theorems.

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.

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