Wright-convexity with respect to arbitrary means

Abstract

Let I ⊂ R be an open interval and M, N :

Decomposition of higher-order Wright convex functions revisited

“…Applying this equality at x + h 0 and at x, and then subtracting the two equalities side by side, we can see that (10)…”

Convexity properties of functions defined on metric Abelian groups

“…The structure and properties of t-convex sets and t-quasiconvex, t-Wright convex, and t-convex functions and their generalizations have been investigated in a large number of recent papers, see e.g. [1,6,7,9,10,[12][13][14][15][16][17][18][19][20][21][22][23]25,[27][28][29][30][31][32][33][34][35]39].…”

“…for some constant constant c, additive function A : X → R and symmetric biadditive function B : X × X → R. Having this form of a, by the additivity of A, the biadditivity and symmetry of B, we can obtaina(x) + a(y) − a(T (x) + (I − T )(y)) − a((I − T )(x) + T (y)) = B(x, x) + B(y, y) − B(T (x) + (I − T )(y), T (x) + (I − T )(y)) − B((I − T )(x) + T (y), (I − T )(x) + T (y)) = 2B(T (x − y), (I − T )(x − y)).Therefore, the function a satisfies the functional Eq. (11)if and only if it has the representation(13) and B(T (x − y), (I − T )(x − y)) = 0 holds for all x, y ∈ X, that is, if condition (12) is valid.…”

Convexity properties of functions defined on metric Abelian groups

“…The structure and properties of t-convex sets and t-quasiconvex, t-Wright convex, and t-convex functions and their generalizations have been investigated in a large number of recent papers, see e.g. [1,6,7,9,10,12,13,14,15,16,17,18,19,20,21,22,23,25,27,28,29,30,31,32,33,34,35,39].…”