Approximate convexity of Takagi type functions

Abstract

Takagi type functionGiven a bounded function Φ : R → R, we define the Takagi type function T Φ : R → R byThe main results of the paper provide sufficient conditions on Φ in order that T Φ be approximately Jensen convex in the following senseApplications to the theory approximately convex functions are also given.

Numerical verification of condition for approximately midconvex functions

“…In this chapter we show a similar result to that from [8] but for continuous piecewise linear approximations of the function α p = x p for p ∈ (0, 1). Example.…”

“…In the last chapter of the paper results similar to those in [8] for continuous piecewise linear functions will be achieved.…”

“…A crucial result in this direction was obtained in 2010 by Makó and Páles [8,Theorem 9] who showed that estimation (1.4) is optimal for the class of functions α(r) = r p for p ∈ (0, 1). In fact they even showed the optimality in the large class of C 1 functions satisfying some additional assumptions.…”

Korovkin type theorems and approximate Hermite–Hadamard inequalities

“…In Remark 3.20, we will deduce this result as a consequence of Corollary 3.19. The following theorem, which has been stated in [17,20,22,27,26], is also a consequence of our main result. We will also derive it in Remark 3.22 by applying Corollary 3.21.…”

Strengthening of strong and approximate convexity∗

“…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].…”