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In scheme theory applied to complex geometry one usually does not encounter coherent rings which are not noetherian as well. However if $X$ is (for example) a Stein manifold then the ring $R = \mathca …

I'm looking for "nice" applications of Liouville's theorem (every bounded entire map is constant) outside the area of complex analysis. An example of what I'm not looking for : a non-constant entire …

No : just take $A = \mathbb{F}_p$. This is the only counterexample : any complete local sub-algebra $\iota_A \colon A \hookrightarrow {\Bbb F}_p[[X]]$, with $A \neq \mathbb{F}_p$, is noetherian of Kr …

The answer is no, even if we assume there are no other poles than $1$ in $\sigma > 1- \epsilon_0$. I give an example below with $\epsilon_0=1$. This is a variant of an example given by Karamata in $19 …

As pointed out by fedja, one can not expect the inequality yo hold with the same epsilon. However, the following holds : if $\mathbf{P}_{z}( |f(z)| \leq \epsilon ) \geq 1 - \delta$, with $\delta \ll n …

It is sufficient to find a non-polynomial entire function $f$ such that $T(r) = \sup_{|z| \leq r} |f(z)|$ satisfies $T^{\circ n}(r) = O_n(e^r)$. Let $r \in \mathbb{R}_+ \mapsto S(r) \in \mathbb{R}^*_+ …

Take $f_n(z) = 3z^{3n} - z^n$ and $C = \{ e^{i\theta} \ | \ \theta \in (0,\pi) \}$. It satisfies all of your hypotheses with $$ f(z) = \frac{2+z}{1+z+z^2} $$ and $\int_C f_n =0$ for $n \geq 1$. But $ …