Muhammad TariqAsif Ali ShaikhSoubhagya Kumar SahooHijaz AhmadThanin SitthiwiratthamJiraporn Reunsumrit2025-03-102025-03-102021Axioms2075168010.3390/axioms100402962-s2.0-85120072863https://repository.dusit.ac.th//handle/123456789/4716The 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 HšlderÕs integral inequality, the improved power-mean integral inequality, and the Hšlder-__can 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. © 2021 by the authors.All Open Access; Gold Open AccessHšlderÕs inequalityImproved power-mean integral inequalityM-preinvex functionPreinvex functionS-type convex functionArticleScopus