Generalizations of Simpson-type inequalities for differentiable functions via $(\alpha ,m)$-convexity

Abstract In the paper, the authors identify and correct several errors and methodological shortcomings in the work of Qaisar, He, and Hussain, “A generalizations of Simpson’s type inequality for differentiable functions using $(\alpha ,m)$ ( α , m ) -convex functions and applications,” J. Inequal. Appl. 2013 , Paper No. 158, 13 pp.; doi: https://doi.org/10.1186/1029-242X-2013-158 . These issues include improperly specified domains, the incorrect conflation of α -convexity with $(\alpha ,m)$ ( α , m ) -convexity, and invalid applications to special means. By presenting explicit counterexamples and establishing corrected theorems under appropriate hypotheses, the authors rectify these mathematical inaccuracies and strengthen the theoretical foundations of research on $(\alpha ,m)$ ( α , m ) -convex functions and their associated integral inequalities.