Numerical verification of condition for approximately midconvex functions

Abstract

Let X be a normed space and V be a convex subset of X. Let α :It can be shown that every continuous α-midconvex function satisfies the following estimation:It is an important problem to verify for which functions α the above estimation is optimal. The conjecture of Páles that this is the case for functions of type α(r) = r p for p ∈ (0, 1), was proved by Makó and Páles (J Math Anal Appl 369:545-554, 2010). In this paper we present a computer assisted method to verify the optimality of this estimation in the class of piecewise linear functions α.Mathematics Subject Classification (2010). Primary 26A51; Secondary 39B62, 65G40.

On approximately monotone and approximately Hölder functions

“…Therefore, for p > 1 a function f : I → R satisfies (1) for some nonnegative ε if and only if f is nondecreasing. Another motivation for our paper comes from the theory of approximate convexity which has a rich literature, see for instance [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [36], [37], [38], [39]. In these papers several aspects of approximate convexity were investigated: stability problems, Bernstein-Doetsch-type theorems, Hermite-Hadamard type inequalities, etc.…”