# Show that the number of zeros of $e^{2z}-P(z)$ is not finite

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.

• Here are two other proofs (not based on Jensen's formula): math.stackexchange.com/q/1782941/42969 Commented Jan 16 at 14:58
• I am aware of those proofs, but I haven't seen Hadamard's Theorem in class, and since the question specifically asks to use Jensen's Formula, I'd like to see a solution using it. Commented Jan 16 at 15:10