On approximately convex Takagi type functions

Abstract:

Given a nonnegative function $\phi :[0,\frac 12]\to \mathbb {R}_+$, we define the Takagi type function $S_\phi :\mathbb {R}\to \mathbb {R}$ by \begin{equation*} S_\phi (x):=\sum _{n=0}^{\infty }2\phi \big (\tfrac {1}{2^{n+1}}\big )d_Z(2^nx), \end{equation*} where $d_{\mathbb {Z}}(x):=\operatorname {dist}(x,\mathbb Z) :=\inf \{|x-k|: k\in \mathbb Z\}.$ The main result of the paper states that if $\phi (0)=0$ and the mapping $x\mapsto \phi (x)/x$ is concave, then the Takagi type function $S_\phi$ is approximately Jensen convex in the following sense: \begin{equation*} S_\phi \Big (\frac {x+y}{2}\Big ) \leq \frac {S_\phi (x)+S_\phi (y)}{2}+\phi \circ d_\mathbb Z\Big (\frac {x-y}{2}\Big ) \qquad (x,y\in \mathbb {R}). \end{equation*} Applications to the theory of approximately convex functions are also given.