G-Majorization and Fejér and Hermite-Hadamard Like Inequalities for G-Symmetrized Convex Functions

For a given finite group G acting on an inner product space V, we introduce G-symmetrized convex functions and G-symmetrized increasing functions. We use them to establish some inequalities of Fejér and Hermite-Hadamard types. In particular, we apply Wright convexity. We interpret the obtained results for the group G = Cn of sign changes on V = Rn. The case n = 1 leads to some recent results by S. S. Dragomir [Symmetrized convexity and Hermite-Hadamard type inequalities, J. Math. Inequalities 10/4 (2016) 901--918] and S. Abramovich and L.-E. Persson [Some new Hermite-Hadamard and Fejer type inequalities without convexity/concavity, Math. Inequalities Appl. 23/2 (2020) 447--458].