It’s a matter of principles (of convergence)

Let (f_n)_{n\in \mathbb{N}} be a sequence of entire functions such that, for every z \in \mathbb{C}, there exists F(z)=sup_{n\in \mathbb{N}}|f_n(z)| \in \mathbb{R}. Suppose that F:\mathbb{C} \to \mathbb{R} is continuous and F(0)=0. Let g be an entire function such that g'(0) \neq 0. Prove that g \circ f_n \to g over compact sets if and only if f_n(z) \to z, for every z \in \mathbb{C}.

Advertisements
It’s a matter of principles (of convergence)

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s