Abstract
The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m–preinvex function and to present inequalities of the Hermite–Hadamard type in the setting of the generalized s-type m–preinvex function. First, we give the definition and then investigate some of its algebraic properties and examples. We also present some refinements of the Hermite–Hadamard-type inequality using Hölder’s integral inequality, the improved power-mean integral inequality, and the Hölder-İşcan integral inequality. Finally, some results for special means are deduced. The results established in this paper can be considered as the generalization of many published results of inequalities and convexity theory.
Highlights
The theory of convex functions has become a rich source of inspiration in different fields of science
Motivated by ongoing research activities, in this article, we focus on introducing a new family of preinvex functions called the generalized s-type m–preinvex function
We recall some known concepts related to preinvex functions, which we use extensively throughout the paper
Summary
The theory of convex functions has become a rich source of inspiration in different fields of science. It has been proven that the notion of the convex function has a very special relationship with the theory of inequalities. Among all the essential integral inequalities, the Hermite–Hadamard inequality holds a special place in the heart of all the mathematicians who work in the field of convexity theory. This inequality was first introduced by Hermite in 1881 in the journal Mathesis (see [6]). Introduced the n–polynomial convex function; explored its properties; and by applying this convexity, presented a novel version of the Hermite–Hadamard-type inequality. The results proved in this paper continue to hold for these special cases
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
