Convexity with respect to families of means

Generalizing Jensen and Bregman divergences with comparative convexity and the statistical Bhattacharyya distances with comparable means

“…We can thus give a complete (P δ1 , P δ2 )-convex characterization of functions f : I ⊂ R ++ → R ++ , see [28]. Define function f δ1,δ2 : I δ1 → R with I δ = {x δ : x ∈ I} for δ = 0 (and I 0 = {log x : x ∈ I}) as:…”

“…A function satisfying this Jensen convexity inequality property may not be continuous [28]. But it turns out that for a continuous function F , the midpoint convexity implies the general convexity definition of Eq.…”

On T-Schur Convex Maps

“…The definition of t-Wright convex functions was introduced by Matkowski in [11]. The connection between t-Wright convexity and Jensen convexity was investigated in [10,15]. In [15] the necessary and sufficient topological conditions under which every t-Wright convex function has to be Jensen convex are given.…”

“…In [15] the necessary and sufficient topological conditions under which every t-Wright convex function has to be Jensen convex are given. In [10] the authors solved an algebraic problem posed by Matkowski in [11], who asked whether a t-Wright convex function with a t ∈ (0, 1) has to be Jensen convex? In [10] Maksa et al gave the positive answer to the problem of Matkowski for all rational t ∈ (0, 1) and certain algebraic values of t. However, they proved that if t is either transcendental or the distance of some of the algebraic (maybe complex) conjugate of t from 1 2 is at least 1 2 , then there exists a function which is t-Wright convex but not Jensen convex.…”