There is a question in "An introduction to harmonic analysis" by Yitzhak Katznelson.
Problem: Let $f\in A(\mathbb T)$($\mathbb T$ denotes the torus), i.e. $\sum\limits_{n\in\mathbb Z}|\hat f(n)|<\infty$, $f$ is real valued and monotonic in a neighborhood of $t_0\in\mathbb T$. Show that $$|f(t)-f(t_0)|=O((\log|t-t_0|^{-1})^{-1}) \text{ as }t\rightarrow t_0.$$
I try to compute the Fourier series of $f(t)-f(t_0)$ directly and use Abel summation. However, it seems to me that monotonic makes nonsense. Can someone tell me how to solve the problem? Many Thanks!