On the stability of t-convex functions

Abstract

A real-valued function f defined on an open convex setfor all x, y ∈ D, where d : X × X → R is a given function and t ∈]0, 1[ is a fixed parameter.The main result of the paper states that if f is locally bounded from above at a point of D and (d, t)-convex then it satisfies the convexity-type inequality (under some assumptions)for all x, y ∈ D and s ∈ [0, 1], where ϕ : [0, 1] → R is defined as the fixed point of a certain contraction. The main result of this paper offers a generalization of the celebrated Bernstein and Doetsch theorem and the recent results by Nikodem and Ng, Páles and the author. (2000). 26A51, 26B25, 39B62. Mathematics Subject Classification

Approximate convexity of Takagi type functions

“…The other concerns, roughly speaking, estimations of the bounds which appear for approximately convex functions, see, for example, Cannarsa and Sinestrari [4], Green [8], Laczkovich [17], Ng and Nikodem [20], Rolewicz [28,29,31]. Our considerations belong to the second current which is motivated by the fact that Takagi-like functions appear naturally in the investigation of approximate convexity, see, for example, Házy [10], Házy and Páles [11][12][13], Makó and Páles [18], Mureńko, Tabor and Tabor [19], Tabor and Tabor [32,33], Tabor, Tabor, anḋ Zołdak [34]. The role and importance of the Takagi function in the theory of approximate convexity was discovered by Házy and Páles [11] who obtained the following result.…”

Bernstein--Doetsch type theorems for set-valued maps of strongly and approximately convex and concave type

“…Results, extending this approach to more general error terms and also to convexity concepts related to Chebyshev systems, have recently been obtained by Házy, Makó and Páles [7,8,10,11,15,16,17,18] and by Mureńko, Ja. Tabor, Jó.…”

“…provided that D is a subset of a normed space X and f is real valued. They showed, under the usual local upper-boundedness condition, that (8) implies…”

Strengthening of strong and approximate convexity∗

“…Obviously, by (5), the notion of (E, t)-convexity is equivalent to (E, 1 − t)-convexity. If (6) holds for all x, y ∈ D with t = 1 2 , then f is called E-Jensen convex or E-midconvex.…”

“…An important issue in these papers is to obtain estimations of the bounds which appear for approximately convex functions, see, for example, Cannarsa and Sinestrari [1], Ng and Nikodem [15], Rolewicz [17][18][19]. It also turned out that Takagi-like functions appear naturally in the investigation of approximate convexity, see, for example, Házy [5], [6], Házy and Páles [7][8][9], Makó and Páles [11][12][13], Mureńko, Tabor and Tabor [14], Tabor and Tabor [20,21], Tabor, Tabor andŻo ldak [22].…”