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Abstract We introduce a new class of functions called strongly ( η , ω ) $(\eta,\omega)$ -convex functions. This class of functions generalizes some recently introduced notions of convexity, namely, the η-convex functions and strongly η-convex functions. We also establish inequalities of the Hermite–Hadamard–Fejér’s type, which generalize results of Delavar and Dragomir (Math. Inequal. Appl. 20(1):203–216, 2017) and Awan et al. (Filomat 31(18):5783–5790, 2017). In addition, we obtain some new results for this class of functions. Finally, we apply our results to the k-Riemann–Liouville fractional integral operators to obtain more results in this direction.