scholarly journals Refined estimates and generalization of some recent results with applications

2021 ◽  
Vol 6 (10) ◽  
pp. 10728-10741
Author(s):  
Aqeel Ahmad Mughal ◽  
◽  
Deeba Afzal ◽  
Thabet Abdeljawad ◽  
Aiman Mukheimer ◽  
...  
Keyword(s):  
Convex Functions ◽  
Type Result ◽  
Linear Spaces ◽  
New Class ◽  
Discrete Inequalities ◽  
Harmonic Convex ◽  
Class Of Functions

<abstract><p>In this paper, we firstly give improvement of Hermite-Hadamard type and Fej$ \acute{e} $r type inequalities. Next, we extend Hermite-Hadamard type and Fej$ \acute{e} $r types inequalities to a new class of functions. Further, we give bounds for newly defined class of functions and finally presents refined estimates of some already proved results. Furthermore, we obtain some new discrete inequalities for univariate harmonic convex functions on linear spaces related to a variant most recently presented by Baloch <italic>et al.</italic> of Jensen-type result that was established by S. S. Dragomir.</p></abstract>

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 Hermite–Hadamard-Type Inequalities for the Generalized Geometrically Strongly Modified h -Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Xishan Yu ◽  
Muhammad Shoaib Saleem ◽  
Shumaila Waheed ◽  
Ilyas Khan
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Applied Mathematics ◽  
Optimization Theory ◽  
New Class ◽  
Convexity Theory ◽  
Class Of Functions

Convexity theory becomes a hot area of research due to its applications in pure and applied mathematics, especially in optimization theory. The aim of this paper is to introduce a broader class of convex functions by unifying geometrically strong convex function with h convex functions. This new class of functions is called as generalized geometrically strongly modified h -convex functions. We established Hermite–Hadamard-type inequalities for the generalized geometrically strongly modified h -convex functions. Our results can be considered as generalization and extension of literature.

Integral inequalities for differentiable p-harmonic convex functions

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6575-6584 ◽  
Author(s):  
Muhammad Noor ◽  
Khalida Noor ◽  
Sabah Iftikhar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Integral Inequalities ◽  
Absolute Value ◽  
New Class ◽  
Special Cases ◽  
Harmonic Convex

In this paper, we consider a new class of harmonic convex functions, which is called p-harmonic convex function. Several new Hermite-Hadamard, midpoint, Trapezoidal and Simpson type inequalities for functions whose derivatives in absolute value are p-harmonic convex are obtained. Some special cases are discussed. The ideas and techniques of this paper may stimulate further research.

scholarly journals Approximately two-dimensional harmonic $(p_{1},h_{1})$-$(p_{2},h_{2})$-convex functions and related integral inequalities

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Saad Ihsan Butt ◽  
Artion Kashuri ◽  
Muhammad Nadeem ◽  
Adnan Aslam ◽  
Wei Gao
Keyword(s):  
Convex Functions ◽  
Integral Inequalities ◽  
Two Dimensional ◽  
New Class ◽  
Special Cases ◽  
Related Integral ◽  
Harmonic Convex

Abstract The aim of this study is to introduce the notion of two-dimensional approximately harmonic $(p_{1},h_{1})$ ( p 1 , h 1 ) -$(p_{2},h_{2})$ ( p 2 , h 2 ) -convex functions. We show that the new class covers many new and known extensions of harmonic convex functions. We formulate several new refinements of Hermite–Hadamard like inequalities involving two-dimensional approximately harmonic $(p_{1},h_{1})$ ( p 1 , h 1 ) -$(p_{2},h_{2})$ ( p 2 , h 2 ) -convex functions. We discuss in detail the special cases that can be deduced from the main results of the paper.

scholarly journals A Class of SemilocalE-Preinvex Functions and Its Applications in Nonlinear Programming

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
Hehua Jiao ◽  
Sanyang Liu ◽  
Xinying Pai
Keyword(s):  
Nonlinear Programming ◽  
Optimality Conditions ◽  
Convex Functions ◽  
Convex Set ◽  
Basic Properties ◽  
New Class ◽  
Duality Results ◽  
Invex Set ◽  
Preinvex Functions ◽  
Class Of Functions

A kind of generalized convex set, called as local star-shapedE-invex set with respect toη,is presented, and some of its important characterizations are derived. Based on this concept, a new class of functions, named as semilocalE-preinvex functions, which is a generalization of semi-E-preinvex functions and semilocalE-convex functions, is introduced. Simultaneously, some of its basic properties are discussed. Furthermore, as its applications, some optimality conditions and duality results are established for a nonlinear programming.

scholarly journals On approximately harmonic h-convex functions depending on a given function

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3783-3793 ◽  
Author(s):  
Muhammad Awan ◽  
Muhammad Noor ◽  
Marcela Mihai ◽  
Khalida Noor ◽  
Nousheen Akhtar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Integral Inequalities ◽  
New Class ◽  
Special Cases ◽  
Harmonic Convex ◽  
Harmonic Convexity

A new class of harmonic convex function depending on given functions which is called as ?approximately harmonic h-convex functions? is introduced. With the discussion of special cases it is shown that this class unifies other classes of approximately harmonic h-convex function. Some associated integral inequalities with these new classes of harmonic convexity are also obtained. Several special cases of the main results are also discussed.

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.

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