Aim A quick search for duplicates. Question If $|f(z) |\leq 1 + |z|$, show that $f(z) = az + b$ is a recently-posted question. I tried to search for its duplicates on Approach0 but I didn't manage to get something like Showing that $f$ is linear function if $\forall z \in \mathbb{C}$, $|f(z)| \l...

Let $f$ be an entire function that satisfies $|f(z)| \leq 1 + |z|$ for all $z\in \mathbb{C}$. Show that $f(z) = az +b$ for fixed complex numbers $a,b$. The hint tells us to try and use Cauchy's Integral Formula on an arbitrary circle. This is my attempt: Consider an arbitrary large circle w...

Approach0 Searching for a specific formula is often rather difficult. A tool which is very suitable for this is the Approach0 search engine. Some basic information can be found in Guide for New Users (which contains also animations of some examples). If you want to know more about the project, ...