Hermite-Hadamard, Jensen’s, Fejér’s, and Simpson’s new inequalities for a merging between two convex functions

Inequalities are widely used to find the optimal solution for optimisation, engineering and linear programming problems. In previous works, only a single convex function such as s-convex, h-convex and g-convex functions were used to construct the inequalities namely Hermite-Hadamard, Jensen’s, Fejér’s, and Simpson’s inequalities. These inequalities use the properties of a single convex function but can only address a limited number of problems. By combining these convex functions, more problems can be solved, therefore their applicability to various fields. Thus, this study aims to construct new inequalities by combining three single convex functions namely sconvex, h-convex and g-convex functions. The approach of combining two or more
classes of convex functions to form hybrid convex function classes aims to extend mathematical inequalities. The findings of the study establish various extensions of the Hermite-Hadamard, Jensen’s, Fejér’s, and Simpson’s inequalities for new classes of functions with hybrid convexity properties. These results are further formulated as
theorems to derive error estimations for the Simpson’s, midpoint, and trapezoidal rules and also to determine the mean value of real numbers. These error estimations are found more accurate compared to the existing inequalities. Within the framework of convexity combinations, this study contributes significantly to the field by providing various extensions of the Hermite-Hadamard, Jensen’s, Fejér’s, and Simpson’s inequalities for hybrid convex functions. In conclusion, this study enhanced mathematical inequalities, optimizing solutions in various fields while preserving convexity properties and extending classical integral approximation methods.