Complex Variables – Zeros of Entire Functions with Parameter

Question

Let $f_w:\mathbb C \to \mathbb C$ be an entire function with $f_w(0)=1$ and at least one root for any choice of $w \in (0,1)$. Assume further that for a dense set of $w$ the function $f_w$ has infinitely many distinct roots and that $w\mapsto f_w$ depends real-analytically on $w.$ Does $f_w$ necessarily have infinitely many distinct roots for all $w \in (0,1)$?

Answer 1

$$f_w(z)=(1-z)e^z+(2w-1)(e^z-1)$$ is a counterexample.

More specifically, the only root of $f_{1/2}$ is $1$, whereas for each $w\in(0,1)\setminus\{1/2\}$ the function $f_w$ has infinitely many roots, of the form $2w+W_k\left(e^{-2 w}(1-2 w)\right)$, where $k$ is any integer and $W_k$ is the $k$th branch of the Lambert $W$ function.