Holomorphic function bounded in a sector with angle $>\pi$ [closed]

Question

I know that according to Liouville’s theorem, if a holomorphic function is bounded on all of C, it is constant. This got me thinking if I could find holomorphic non-constant functions that are bounded in a sector. The first obvious example would be the exponential function which is bounded on the half plane where the real part is non positive. But I cannot think of any entire function that is bounded on a sector with angle $>\pi$. Can anyone find such a function? Or perhaps provide a proof that no such functions exists.

Answer 1

Let $L$ be the boundary of the strip $$ G := \{a + ib\colon a>0, -\pi < b < \pi\}, $$ parameterised in clockwise direction. Define $$ F_0\colon \mathbb{C}\setminus \overline{G}\to\mathbb{C}; \quad z\mapsto \frac{1}{2\pi i}\int_L\frac{\exp(e^t)}{t-z}{\rm{d}}t.$$ It is easy to see that the integral converges and is uniformly bounded (in fact tends to zero as $z\to\infty$). Furthermore, $F_0$ extends to an entire function of the complex plane, which has the desired property. See Gwyneth M. Stallard, The Hausdorff dimension of Julia sets of entire functions, Ergodic Theory Dynam. Systems 11(1991), no. 4, 769–777.

In a similar manner, Cauchy integrals can be used to construct entire functions that are bounded outside any given unbounded Jordan domain. See e.g. https://arxiv.org/abs/1106.3439 .