I'm trying to solve the following problem:
Let $P(z) \neq 0$ be a complex polynomial. Use Jensen's Formula to show that the set of zeros of $e^{2z}-P(z)$ is not finite.
ATTEMPT
Suppose that it were finite, say the zeros are $\alpha_1,\dots,\alpha_n$, and they are inside of $D(0,R)$. Then, by Jensen's Formula,
\begin{align*} \log |f(0)|=-\sum_{k=1}^n \log \left(\frac{R}{\left|a_k\right|}\right)+\frac{1}{2 \pi} \int_0^{2 \pi} \log \left|f\left(R e^{i \theta}\right)\right| d \theta, \end{align*}
where $f(z)= e^{2z}-P(z)$.
Now my idea was to show that $\int_0^{2 \pi} \log \left|f\left(R e^{i \theta}\right)\right| d \theta$ is divergent, and therefore the formula wasn't valid, I'm not sure how to compute $\log \left|e^{Re^{i\theta}} -P(Re^{i\theta})\right|$. I'm also aware of the following inequality derived from Jensen's Formula:
\begin{align*} n(R)\leq CR^{\rho}, \end{align*}
where $\rho$ is the order of growth and $n(R)$ is the number of zeros inside $D(0,R)$, but it doesn't seem relevant here since it's a $\leq$-type inequality.