Aequationes Math. 74 (2007) 210–218 0001-9054/07/030210-9 DOI 10.1007/s00010-007-2880-z c  Birkh¨ auser Verlag, Basel, 2007 Aequationes Mathematicae On the stability of t-convex functions Attila H´ azy Summary. A real-valued function f defined on an open convex set D ⊆ X is called (d, t)-convex if it satisfies f (tx + (1 - t)y) ≤ tf (x) + (1 - t)f (y)+ d(x, y) for 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) f ( sx + (1 - s)y ) ≤ sf (x) + (1 - s)f (y)+ ϕ(s)d(x, y) 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, P´ales and the author. Mathematics Subject Classification (2000). 26A51, 26B25, 39B62. Keywords. Convexity, approximately convexity, (d, t)-convexity, Bernstein–Doetsch theorem. 1. Introduction Let (X, |·|) be a normed space and D ⊆ X be a nonempty open convex set throughout this paper. We investigate the following generalization of the convexity property. Given a function d : X × X → R and t ∈]0, 1[, a function f : D → R is said to be (d, t)-convex if f (tx + (1 - t)y) ≤ tf (x) + (1 - t)f (y)+ d(x, y) for all x, y ∈ D. Due to symmetry, we may assume that 0 <t ≤ 1/2 in sequel. Some particular cases were investigated in several papers. The particular case d(x, y)= k ∑ i=0 ε i |x -y| pi was studied by H´azy and P´ales in [4]: This research was supported by the Hungarian Scientific Research Fund (OTKA) Grant T-038072 and K-62316.