For $n \geq 1,$ let $M_n = n! (\log n)^n$ and set $M_0=1.$ Say that an infinitely differentiable real function $f$ belongs to the quasi-analytic class $C\{M_n\}$if there are constants $\beta_f,B_f$ depending only on $f,$ such that $$||D^n f||_\infty \leq \beta_f (B_f)^n M_n \qquad \forall n \geq 0,$$
where $||-||_\infty$ is the supremum norm on $\mathbb{R}.$
How can I find a function $f$ belonging to $C\{M_n\}$ that is not analytic in an open set of $\mathbb{C}$ containing $\mathbb{R}?$
The classic construction due to Borel are functions of the form $$f(z)=\sum_{n=1}^{\infty}\frac{A_n}{z-z_n}$$ with the $z_n\in\mathbb{C}\setminus\mathbb{R}$ tending to a real, say $z_n\to0$.
Here you can find one way to define $A_n,z_n$ such that $f(z)$ is not analytic at $z=0$, in particular it will not be analytic in a neighborhood of the real axis.