Approximate convexity of Takagi type functions

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