Let $\phi : \mathbb{C} \to \mathbb{C}$ be an entire function satisfying the following three properties:
- $|\phi'(z)| \leq |\phi(z)|$ for all $z \in \mathbb{C}$
- $\phi(0) = 2$
- $\phi(1) = 1$
The problem I'm working on says to show that (a) $\phi$ never vanishes, and (b) that $\phi$ is uniquely determined by these properties. I was able to prove (a) using the Argument Principle, and was able to find a function that satisfied all these properties, namely $\phi(z) = 2 e^{- (\ln 2) z}$. I'm stuck on how to prove that this is the only solution, and would appreciate help, because I'm at a loss.
Consider function $f(z) = \frac{\phi'(z)}{\phi(z)}$. $|f(z)| \leq 1$ and $f$ is entire - thus constant. So we have $\phi'(z) = a\cdot \phi(z)$, so $\phi(z) = b \cdot \exp(a \cdot z)$. And initial conditions are enough to find $a$ and $b$.