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At least a variant of your statement is true, see Lemma 3.2 in my paper with Miriam Benini. Let $f$ is a transcendental entire function that is bounded on an unbounded connected set. (This is always t …

Here's a slightly more hands-on point of view, using just the Jordan curve theorem (although of course in the end it does come down to the same thing as Euler's formula somehow, as described by Alex.) …

I hope that the following result by Bishop and myself might be "interesting": Every open Riemann surface is a cover over the sphere, branched over only three points. Equivalently, every open Riemann s …

Here is one possible elementary argument (somewhat inspired by my paper with Shen in the Monthly, "The exponential map is chaotic"), which avoids any mention of the classification of Fatou components, …

The paper of Detlef Bargmann, "Normal families of covering maps" in Journal d'Analyse, 2001, is relevant to this question; see http://dx.doi.org/10.1007/BF02788084 . His Theorem 1 implies, in particul …

Here is an attempt to construct an example. I am going to let $K_1$ and $K_2$ be compact subsets in the sphere $\hat{\mathbb{C}}$, rather than the plane (of course, we can change coordinates to move i …

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\m …

It depends on what you mean by precise coordinates. I am not sure that I would expect to find a number that has a specific closed form. But then, I do not know how to find a "precise" point where the …

I started to write this as comments, but really it is too long for a comment. It is something of an elaboration on Alex's answer. The theory is analogous to the theory of the boundary behaviour of con …

The measure of maximal entropy is the unique measure that is "fully invariant" in your sense. I believe that this already follows from the original proofs - indeed, it is well-known that if you take a …

Yes, you can say things like this. The easiest way to get these kind of results is probably the Gehring-Hayman theorem. It states that, for two points $z$ and $w$ in a simply-connected domain $D$ or …

I am not sure that I understand your question correctly. In particular, I am not sure what $+\infty$ and $-\infty$ mean. As I first understood it, the answer is negative. Let $A(r,1)$ be the annulus …

As stated, the answer is negative. I will provide a sketch. Indeed, consider some non-star-like Jordan domain $V$, and a conformal isomorphism $\phi\colon V\to\mathbb{D}$ from $V$ to the unit disc. T …

To elaborate on my comment, choose three pairwise disjoint Jofrdan $U_1$, $U_2$, $U_3$ whose boundary passes through infinity, and let $\gamma_i\subset U_i$ be a curve to infinity for each $i$. Fix $\ …

To put the existing answer in context, you will never get summability if your function is holomorphic at $0$ and $f'(0)=1$. In particular, the example given in your question is actually also not summa …