SU\G(š”ø)/KĀ·U
Sign Up for free Log In

Extension of Hermite-Hadamard Type Integral Inequality Whose Second Order Derivatives are m- Convex Functions

Authors

Pitamber Tiwari, Chet Raj Bhatta
Type: Article
Publication Date: 2025-04-21
Citations: 0
DOI: https://doi.org/10.3126/njmathsci.v6i1.77377

Abstract

Integral inequality is a fascinating research domain that helps to estimate the integral mean of convex functions. The convexity theory plays a basic role in the development of various branches of applied sciences. Convexity and inequality are connected which has a fundamental character in many branches of pure and applied disciplines. The Hermite-Hadamard (H-H) type integral inequality is one of the most important inequalities associated with the convex functions. The researchers are being motivated to the extensions, enhancements and generalizations of H-H type inequality for different types of convex functions. In this paper, we have obtained an extension of some integral inequalities of Hermite-Hadamard type for m-convex functions with second order derivatives on the basis of the classical convex functions.

Topics

Locations

  • Nepal Journal of Mathematical Sciences
    View

Hermite-Hadamart Integral Inequality for Differentiable m- Convex Functions

2024-07-02
Pitambar Tiwari , Chet Raj Bhatta
Convexity in connection with integral inequalities is an interesting research domain in recent years. The convexity theory plays a fundamental role in the development of various branches of applied sciences ā€¦ Convexity in connection with integral inequalities is an interesting research domain in recent years. The convexity theory plays a fundamental role in the development of various branches of applied sciences since it includes the theory of convex functions that possesses two important attributes viz. a boundary point is where the maximum value is reached and any local minimum value is a global one. Convexities and inequalities are connected which has a basic character in many branches of pure and applied disciplines. The most important inequality related to convex function is the Hermite-Hadamard integral inequality. The extensions, enhancements and generalizations of this inequality has motivated the researchers in recent years. This paper is an extension of some inequalities connected with difference of the left-hand part as well as the right-hand part from the integral mean in Hermite- Hadamardā€™s inequality for the case of m- convex functions.
View PDF Chat

(m1,m2)-Convexity and Some New Hermite-Hadamard Type Inequalities

2019-12-01
Huriye Kadakal
In this manuscript, a new class of extended (m1,m2)-convex and concave functions is introduced. After some properties of (m1,m2)-convex functions have been given, the inequalities obtained with Holder and Holder-Iscan ā€¦ In this manuscript, a new class of extended (m1,m2)-convex and concave functions is introduced. After some properties of (m1,m2)-convex functions have been given, the inequalities obtained with Holder and Holder-Iscan and power-mean and improwed power-mean integral inequalities have been compared and it has been shown that the inequality with Holder-Iscan inequality gives a better approach than with Holder integral inequality and improwed power-mean inequality gives a better approach than with power-mean inequality.

Novel Analysis of Hermiteā€“Hadamard Type Integral Inequalities via Generalized Exponential Type m-Convex Functions

2021-12-22
Muhammad Tariq , Hijaz Ahmad , Clemente Cesarano +3 more
The theory of convexity has a rich and paramount history and has been the interest of intense research for longer than a century in mathematics. It has not just fascinating ā€¦ The theory of convexity has a rich and paramount history and has been the interest of intense research for longer than a century in mathematics. It has not just fascinating and profound outcomes in different branches of engineering and mathematical sciences, it also has plenty of uses because of its geometrical interpretation and definition. It also provides numerical quadrature rules and tools for researchers to tackle and solve a wide class of related and unrelated problems. The main focus of this paper is to introduce and explore the concept of a new family of convex functions namely generalized exponential type m-convex functions. Further, to upgrade its numerical significance, we present some of its algebraic properties. Using the newly introduced definition, we investigate the novel version of Hermiteā€“Hadamard type integral inequality. Furthermore, we establish some integral identities, and employing these identities, we present several new Hermiteā€“Hadamard Hā€“H type integral inequalities for generalized exponential type m-convex functions. These new results yield some generalizations of the prior results in the literature.
View PDF Chat

A generalization of midpoint inequality of m-convex functions

2016-01-01
Mehmet EyĆ¼p KiĢ‡riĢ‡ÅŸ , Naki Ƈaltıner
In this paper, some new integral inequalities will be given using generalized Hermite-Hadamard's type integral inequalities holding for mā€“convex functions. Our results presented here would provide extensions of those given ā€¦ In this paper, some new integral inequalities will be given using generalized Hermite-Hadamard's type integral inequalities holding for mā€“convex functions. Our results presented here would provide extensions of those given in earlier works.

Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

2020-09-09
Pshtiwan Othman Mohammed , Thabet Abdeljawad , Shengda Zeng +1 more
Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional ā€¦ Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of Ļ†-Riemannā€“Liouville fractional integral operators and the recently defined class of convex functions, namely the ĻƒĖ˜-convex functions. Based on these, we will obtain few integral inequalities of Hermiteā€“Hadamardā€™s type for a ĻƒĖ˜-convex function with respect to an increasing function involving the Ļ†-Riemannā€“Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemannā€“Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out.
View PDF Chat

Some Integral Inequalities Involving Exponential Type Convex Functions and Applications

2021-12-08
Muhammad Tariq , Hijaz Ahmad , Soubhagya Kumar Sahoo +1 more
In this present case, we focus and explore the idea of a new family of convex function namely exponentialtype mā€“convex functions. To support this newly introduced idea, we elaborate some ā€¦ In this present case, we focus and explore the idea of a new family of convex function namely exponentialtype mā€“convex functions. To support this newly introduced idea, we elaborate some of its nice algebraicproperties. Employing this, we investigate the novel version of Hermiteā€“Hadamard type integral inequality.Furthermore, to enhance the paper, we present several new refinements of Hermiteā€“Hadamard (Hāˆ’H) inequality.Further, in the manner of this newly introduced idea, we investigate some applications of specialmeans. These new results yield us some generalizations of the prior results in the literature. We believe, themethodology investigated in this paper will further inspire intrigued researchers.
View PDF Chat

On Modified Integral Inequalities for a Generalized Class of Convexity and Applications

2023-02-05
H. M. Srivastava , Muhammad Tariq , Pshtiwan Othman Mohammed +3 more
In this paper, we concentrate on and investigate the idea of a novel family of modified p-convex functions. We elaborate on some of this newly proposed ideaā€™s attractive algebraic characteristics ā€¦ In this paper, we concentrate on and investigate the idea of a novel family of modified p-convex functions. We elaborate on some of this newly proposed ideaā€™s attractive algebraic characteristics to support it. This is used to study some novel integral inequalities in the frame of the Hermiteā€“Hadamard type. A unique equality is established for differentiable mappings. The Hermiteā€“Hadamard inequality is extended and estimated in a number of new ways with the help of this equality to strengthen the findings. Finally, we investigate and explore some applications for some special functions. We think the approach examined in this work will further pique the interest of curious researchers.
View PDF Chat

Hermiteā€“Hadamard-Type Inequalities for h-Convex Functions Involving New Fractional Integral Operators with Exponential Kernel

2022-06-01
Yaoā€Qun Wu
In this paper, we use two new fractional integral operators with exponential kernel about the midpoint of the interval to construct some Hermiteā€“Hadamard type fractional integral inequalities for h-convex functions. ā€¦ In this paper, we use two new fractional integral operators with exponential kernel about the midpoint of the interval to construct some Hermiteā€“Hadamard type fractional integral inequalities for h-convex functions. Taking two integral identities about the first and second derivatives of the function as auxiliary functions, the main results are obtained by using the properties of h-convexity and the module. In order to illustrate the application of the results, we propose four examples and plot function images to intuitively present the meaning of the inequalities in the main results, and we verify the correctness of the conclusion. This study further expands the generalization of Hermiteā€“Hadamard-type inequalities and provides some research references for the study of Hermiteā€“Hadamard-type inequalities.
View PDF Chat

On Some New Integral Inequalities Related With The Hermite-Hadamard Inequality via h-Convex Functions

2017-07-28
Jorge E. HernƔndez HernƔndez
In this paper, some new integral inequalities related with the Hermite-Hadamard inequality, using generalizations of convex functions, in particular, h-convex functions, and from which it is possible to generalize other ā€¦ In this paper, some new integral inequalities related with the Hermite-Hadamard inequality, usingĀ  generalizations of convex functions, in particular, h-convex functions, and from which it is possible to generalize other results referring to s-convex functions and P-convex functions.

General Convexity of Multidimensional Functions and Related Hermite-Hadamard Type Integral Inequalities

2019-12-31
Fatma Buğlem YALƇIN , NurgĆ¼l Okur Bekar
The basic goal is to investigate general convexity of multidimensional functions and derive several important inequalities associated with itā€™s in this paper. For this reason, multidimensional general convex functions were ā€¦ The basic goal is to investigate general convexity of multidimensional functions and derive several important inequalities associated with itā€™s in this paper. For this reason, multidimensional general convex functions were firstly defined. Afterwards, some properties of these functions were mentioned. Accordingly, the relation of multidimensional general convex functions with other convex functions was established. Additionally, a generalization of Hermite-Hadamard type integral inequality was showed for two-dimensional general convex functions. Finally , Hermite-Hadamard type integral inequality for multidimensional general convex functions was verified and an explanatory example for this inequality was given in this study.
View PDF Chat

Hermite-Hadamard and Simpsonā€™s type of inequalities for first and second ordered derivatives using product of (h1, h2, s)-convex and m-convex function

2023-01-01
Sabir Yasin , Zurni Omar , Masnita Misiran
This article is dedicated to find the extensions for Hermite-Hadamard (H-H) and Simpsonā€™s type of inequalities. By combining multiple existing convex functions by placing specific restrictions on them is the ā€¦ This article is dedicated to find the extensions for Hermite-Hadamard (H-H) and Simpsonā€™s type of inequalities. By combining multiple existing convex functions by placing specific restrictions on them is the most effective in many approaches to find a new convex function. Here, to find the new function (h1, h2, s)-Convex and m-Convex Function are used. Because of the product of two or even more convex functions does not necessarily have to be convex, we decided to investigate merging distinct convex functions. Combining more than two convex functions in a novel adaptive way advances to new applications in a range of disciplines, including mathematics and other fields. In this paper, some extensions for Hermite-Hadamard and Simpsonā€™s inequalities is explored. The newly constructed extensions of these inequalities will be considered as the improvements and refinements of previously obtained results.
View PDF Chat

Some Results on Hermite-Hadamard Type Inequality through Convexity

2014-04-18
Shahid Qaisar , Sabir Hussain
Our aim in this article to establish various inequalities for some differentiable mapping that are connected with illustrious Hermite-Hadamard integral inequality for mapping whose absolute values of derivatives are convex. ā€¦ Our aim in this article to establish various inequalities for some differentiable mapping that are connected with illustrious Hermite-Hadamard integral inequality for mapping whose absolute values of derivatives are convex. The new integral inequalities are then applied to some special means and as well as numerical integration to obtain some better estimates.
View PDF Chat

INTEGRAL INEQUALITIES OF HERMITE-HADAMARD TYPE FOR ((Ī±;m); log)-CONVEX FUNCTIONS ON CO-ORDINATES

2015-12-01
Bo-Yan Xi , Feng Qi
The convexity of functions is a basic concept in mathematics and it has been generalized in various directions.Establishing integral inequalities of Hermite -Hadamard type for various convex functions is one ā€¦ The convexity of functions is a basic concept in mathematics and it has been generalized in various directions.Establishing integral inequalities of Hermite -Hadamard type for various convex functions is one of the main topics in the theory of convex functions and attracts a number of mathematicians for several centuries.Currently an amount of literature on integral inequalities of Hermite -Hadamard type for various convex functions has been accumulated.In the paper the authors introduce a new concept "((Ī±, m), log)-convex functions on the co-ordinates on the rectangle of the plane" and establish new integral inequalities of the Hermite -Hadamard type for ((Ī±, m), log)-convex functions on the co-ordinates on the rectangle of the plane.

The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results

2011-12-28
Erhan Set , Mehmet Zeki Sarıkaya , M. Emіn Ɩzdemіr +1 more
In this paper, firstly we have established Hermite-Hadamard's inequalities for s-convex functions in the second sense and m-convex functions via fractional integrals. Secondly, a Hadamard type integral inequality for the ā€¦ In this paper, firstly we have established Hermite-Hadamard's inequalities for s-convex functions in the second sense and m-convex functions via fractional integrals. Secondly, a Hadamard type integral inequality for the fractional integrals are obtained and these result have some relationships with KBOP.

The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results

2011-01-01
Erhan Set , Mehmet Zeki Sarıkaya , Murat Ɩzdemir +1 more
In this paper, firstly we have established Hermite-Hadamard's inequalities for s-convex functions in the second sense and m-convex functions via fractional integrals. Secondly, a Hadamard type integral inequality for the ā€¦ In this paper, firstly we have established Hermite-Hadamard's inequalities for s-convex functions in the second sense and m-convex functions via fractional integrals. Secondly, a Hadamard type integral inequality for the fractional integrals are obtained and these result have some relationships with KBOP.
View PDF Chat

Some New Mathematical Integral Inequalities Pertaining to Generalized Harmonic Convexity with Applications

2022-09-10
Muhammad Tariq , Soubhagya Kumar Sahoo , Sotiris K. Ntouyas +3 more
The subject of convex analysis and integral inequalities represents a comprehensive and absorbing field of research within the field of mathematical interpretation. In recent times, the strategies of convex theory ā€¦ The subject of convex analysis and integral inequalities represents a comprehensive and absorbing field of research within the field of mathematical interpretation. In recent times, the strategies of convex theory and integral inequalities have become the subject of intensive research at historical and contemporary times because of their applications in various branches of sciences. In this work, we reveal the idea of a new version of generalized harmonic convexity i.e., an mā€“polynomial pā€“harmonic sā€“type convex function. We discuss this new idea by employing some examples and demonstrating some interesting algebraic properties. Furthermore, this work leads us to establish some new generalized Hermiteā€“Hadamard- and generalized Ostrowski-type integral identities. Additionally, employing Hƶlderā€™s inequality and the power-mean inequality, we present some refinements of the Hā€“H (Hermiteā€“Hadamard) inequality and Ostrowski inequalities. Finally, we investigate some applications to special means involving the established results. These new results yield us some generalizations of the prior results in the literature. We believe that the methodology and concept examined in this paper will further inspire interested researchers.
View PDF Chat

Hermite-Hadamard Inequalities for a New Class of Generalized-(s,m) via Fractional Integral

2024-07-08
Erdal GĆ¼l , Ahmet Ocak AkdemiĢ‡r , AbdĆ¼llatif YalƧın
This paper defines a new generalized (s,m)-Ļƒ convex function using the Ļƒ convex functions and provides some applications and exact results for this kind of functions. The new definition of ā€¦ This paper defines a new generalized (s,m)-Ļƒ convex function using the Ļƒ convex functions and provides some applications and exact results for this kind of functions. The new definition of the (s,m)-Ļƒ convex function class is used to obtain the Hermite Hadamard type integral inequalities existing in the literature, and new integral inequalities are obtained with the help of the Ļƒ-Riemann-Liouville fractional integral. Additionally, a new Hermite-Hadamard type fractional integral inequality is constructed using the Ļƒ-Riemann-Liouville fractional integral.
View PDF Chat

Some Hadamard-Type Integral Inequalities Involving Modified Harmonic Exponential Type Convexity

2023-05-05
Asif Ali Shaikh , Evren HınƧal , Sotiris K. Ntouyas +2 more
The term convexity and theory of inequalities is an enormous and intriguing domain of research in the realm of mathematical comprehension. Due to its applications in multiple areas of science, ā€¦ The term convexity and theory of inequalities is an enormous and intriguing domain of research in the realm of mathematical comprehension. Due to its applications in multiple areas of science, the theory of convexity and inequalities have recently attracted a lot of attention from historians and modern researchers. This article explores the concept of a new group of modified harmonic exponential s-convex functions. Some of its significant algebraic properties are elegantly elaborated to maintain the newly described idea. A new sort of Hermiteā€“Hadamard-type integral inequality using this new concept of the function is investigated. In addition, several new estimates of Hermiteā€“Hadamard inequality are presented to improve the study. These new results illustrate some generalizations of prior findings in the literature.
View PDF Chat

(alpha,m1,m2)-convexity and some inequalities of Hermite-Hadamard type

2019-07-06
Huriye Kadakal
In this paper, we introduce a new class of extended (alpha;m1;m2)- convex functions. Some algebraic properties of these class functions have been investigated. Some new Hermite-Hadamard type inequalities are derived. ā€¦ In this paper, we introduce a new class of extended (alpha;m1;m2)- convex functions. Some algebraic properties of these class functions have been investigated. Some new Hermite-Hadamard type inequalities are derived. Results represent signicant refinement and improvement of the previous results. Also, the author establish a new integral identity and, by this identity, Hƶlder's and power mean inequality, discover some new Hermite-Hadamard type inequalities for functions whose first derivatives are (alpha;m1;m2)-convex. Our results are new and coincide with the previous results in special cases.
View PDF Chat

On a new class of convex functions and integral inequalities

2019-05-09
Shanhe Wu , Muhammad Uzair Awan , Muhammad Aslam Noor +2 more
The aim of this paper is to introduce a new extension of convexity called Ļƒ-convexity. We show that the class of Ļƒ-convex functions includes several other classes of convex functions. ā€¦ The aim of this paper is to introduce a new extension of convexity called Ļƒ-convexity. We show that the class of Ļƒ-convex functions includes several other classes of convex functions. Some new integral inequalities of Hermiteā€“Hadamard type are established to illustrate the applications of Ļƒ-convex functions.
View PDF Chat