Let $f$ be an entire function normalized by the conditions $f(0)=0$ and $f'(0)=1$. Find necessary and sufficient conditions on $f$ for normality of the family of successive iterates $\{f\circ f,f\circ f\circ f, f\circ f\circ f\circ f,\ldots \}$. I know that if $|f'(0)|>1$, it certainly does not. But for $|f'(0)|=1$ I’m having trouble coming with conditions […]

Let $\mathcal{F}$ be a family of analytic functions in a domain $G$. Denote by $\mathcal{F}’:=\{ f’ \mid f \in \mathcal{F}\}$. It does not suffice to know that $\mathcal{F}’$ is normal to ensure that $\mathcal{F}$ is normal (namely take $\mathcal{F} = \{ n \mid n \in \mathbb{N}\}$ as a counterexample) but someone assured me that if […]

I’m having difficulty with the following exercise in Ahlfors’ text, on page 227. Prove that in any region $\Omega$ the family of analytic functions with positive real part is normal. Under what added condition is it locally bounded? Hint: Consider the functions $e^{-f}$. Here is what I’ve tried: I will start with a remark: Apparently, […]

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