Define the "small triangle" and "small ball" to the right of $0\in \mathbb C$, \begin{align} T(\epsilon,k) = \{ z \in \mathbb C: |\arg z|<\epsilon , 0< \Re z < k\} \\ C(\epsilon,k) = \{z\in \mathbb C: |z-\epsilon|<\epsilon, 0< \Re z < k\} \end{align} Does there exist a complex function that is:
- analytic and bounded on some $T(\epsilon,k)$,
- not analytic on any $C(\epsilon,k)$, and
- cannot be analytically continued to a proper neighbourhood of $0$?
Remarks
- I want to ignore the counterexamples like $f(z) = z $ for $\arg < c$ and $f(z) = 0$ for $\arg \ge c$.
- Unlike smooth functions, my toolbox of anayltic functions is rather "sparse" without access to mollifiers, and I am not sure what sort of conditions I can enforce. Most of the analytic functions I know that are not entire are either not defined at some poles(meromorphic) or cannot be extended past a ball (lacunary series). With mobius transforms, this gives me examples of functions only analytic on a half plane.
- I know of functions that are analytic and bounded only on a 'sectorial neighbourhood of infinity', e.g. $\exp(-z^4)$, since $\exp(-z^4) \to \infty$ along the rays $$\{\Re z > 0, \arg z = \pm \pi/4\},$$ but if $|\arg z| < \epsilon \ll 1$, then $\cos(4\arg z)> c > 0$, so that $$ |\exp(-z^4)| = \exp(\Re(-z^4) ) = \exp(-|z|\cos(4\arg z)) ≤ \exp(-c|z|) \to 0 $$ as $|z| \to \infty $ in the sector $\{ |\arg z| < \epsilon , \Re z > 0\}$. However, I am looking for a function that exhibits this behavior only a "finite distance away from the singular point".
- The function $z^{s}$ for $s\in(0,1)$ is a bounded function whose derivative is $O(1/|z|^{1-s})$ as $|z|\to 0$ from the right half plane. In fact, for any function $f$ analytic and bounded on $T(\epsilon,k)$, for any $\epsilon' < \epsilon$, we have the Cauchy type estimate, $$\sup_{z\in T(\epsilon',k)} |z f'(z)| \leq C\frac{\sup_{z\in T(\epsilon,k)} |f(z)|}{\epsilon' - \epsilon}.$$ I'm wondering if the functions $z^s$ are the "closest" I can get to achieving equality in this inequality. I am also considering if the angle is only to allow the above quantitative bound for functions that depends on $\epsilon,\epsilon'$, or if there are 'nasty' analytic functions that are only nice on a small triangle $T(\epsilon,k)$.