Let's consider a function of a complex variable $f(z)$, with the following properties:
- $f(z)$ is bounded by a constant as $z \to \infty$ away from the positive real axis ($z \notin \mathbb{R}^+$)
- $f(z)$ is analytic in the interior of the domain $\mathcal{D} = \{ z \in \mathbb{C} : -\epsilon<\arg z < \epsilon \} $, for some finite positive $\epsilon$.
Can one conclude that $f(z)$ is bounded by the same constant also on the positive real axis?
Arguments like the Phragmén–Lindelöf principle get close, but they require some bound on the positive real axis. However, here we know that the function is bounded everywhere, not just on the boundary of a sector, and the boundaries of the sector can get arbitrarily close to the positive real axis.
What I have in mind, specifically, is a function which is analytic away from the negative real axis, and for which I can prove a bound of the following type:
$$z=r e^{\mathbb{i} \theta}, \qquad \lim_{r \to \infty} \left| f(z) \right|(1-\cos\theta)^\alpha <+\infty, $$ for some finite positive $\alpha$.
Thanks a lot!