Approximate convexity of Takagi type functions

Given a bounded function Φ:R→RΦ:R→R, we define the Takagi type function TΦ:R→RTΦ:R→R byTΦ(x):=∑n=0∞Φ(2nx)2n. The main results of the paper provide sufficient conditions on Φ   in order that TΦTΦ be approximately Jensen convex in the following senseTΦ(x+y2)⩽TΦ(x)+TΦ(y)2+Φ(x−y2)−Φ(0)(x,y∈R). Applications to the theory approximately convex functions are also given.