Approximately Midconvex Functions

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.…”

On $\varphi$-convexity

“…Another, but related, notion of approximate convexity, the concept of so-called paraconvexity was introduced by Rolewicz [33,34,35] in the late 70s. It also turned out that Takagi-like functions appear naturally in the investigation of approximate convexity, see, for example, Boros [3], Házy [15,16], Házy and Páles [17,18,19], Makó and Páles [24,25], Mrowiec, Tabor and Tabor [27], Tabor and Tabor [36,37], Tabor, Tabor, and Żołdak [39,38].…”

“…(tx + (1 − t)y) = f y + (2tx + (1 − 2t)y) 2 ≤ f (y) + f (2tx + (1 − 2t)y) 2 + ϕ(t x − y ).On the other hand, by(27), we get thatf (2tx + (1 − 2t)y) ≤ 2tf (x) + (1 − 2t)f (y) + K x,y 2 n + n−1 j=0 ϕ d Z (2 j+1 t) x − y 2 j .Combining these two inequalities, we obtainf (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y) + 1 (2 j+1 t) x − y 2 j + ϕ(t x − y ) = tf (x) + (1 − t)f (y) + K x,y 2 n+1 + n j=0 ϕ d Z (2 j t) x − y 2 j .…”