On approximately convex Takagi type functions (Q2838963)

The celebrated stability result of Hyers and Ulam states that an approximately convex function, that is, a function satisfying the convexity inequality with an error term, is close to a convex function. The investigations and results of this paper are motivated by the fact that Takagi type functions appear naturally in the theory of approximate convexity. The main result of the paper reads as follows:NEWLINENEWLINE{ Theorem.} Define the function \(d_{\mathbb Z}:\mathbb R\to\mathbb R\) by \(d_{\mathbb Z}(x):=\min\{|x-z|\mid z\in\mathbb Z\}\). Let \(\phi: [0,1/2]\to\mathbb R\) be a nonnegative function and consider the Takagi type function \(S_{\phi}:\mathbb R\to\mathbb R\) by NEWLINE\[NEWLINE S_{\phi}:=\sum_{n=0}^{\infty}2\phi\left(\frac{1}{2^{n+1}}\right)d_{\mathbb Z}(x). NEWLINE\]NEWLINE If \(\phi(0)=0\) and the mapping \(x\longmapsto\phi(x)/x\) is concave, then the Takagi type function \(S_{\phi}\) is approximately Jensen convex in the following sense: NEWLINE\[NEWLINE S_{\phi}\left(\frac{x+y}{2}\right)\leq\frac{S_{\phi}(x)+S_{\phi}(y)}{2}+ \phi\circ d_{\mathbb Z}\left(\frac{x-y}{2}\right)\qquad(x,y\in\mathbb R). NEWLINE\]NEWLINENEWLINENEWLINEThe introduction of the paper gives a brief and comprehensive summary (with interesting historical comments) on the most important developments of the theory of approximate convexity and Takagi type functions, as well.