Abstract
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 Hermite–Hadamard type integral inequality. Furthermore, we establish some integral identities, and employing these identities, we present several new Hermite–Hadamard H–H type integral inequalities for generalized exponential type m-convex functions. These new results yield some generalizations of the prior results in the literature.
Highlights
The concept of convexity and its generalization is a rapidly growing area of mathematics, with numerous applications in machine learning, optimization, engineering, management, control theory, economics, and other disciplines
It is especially important in the study of optimization problems, where it is distinguished by many convenient properties
Motivated by the ongoing research activities, this paper aims to introduce a new class of exponential type convex function, called generalized exponential type m-convex function
Summary
The concept of convexity and its generalization is a rapidly growing area of mathematics, with numerous applications in machine (deep) learning, optimization, engineering, management, control theory, economics, and other disciplines. During the last few decades, numerous scientists especially mathematicians have added to the advancement of this beautiful concept in different directions It is especially important in the study of optimization problems, where it is distinguished by many convenient properties (for example, any minimum of a convex function is a global minimum, or the maximum is attained at a boundary point). For the attention of the readers, we encourage the references [1,2,3,4,5,6] It has a strong connection and plays a significant role in the development of the theory of inequalities. During the last twenty years, numerous mathematicians and researchers thought of their incredible commitments and considerations to investigate the Hermite–Hadamard inequality In this manner, a significant amount of literature can be found for the researchers. A new version of Hermite–Hadamard inequality and its refinements are investigated employing this new convexity
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