# A simple question about the zeros of an Entire Function in LP-class

We know that functions in $\mathcal{LP}$-class, and only these, are uniform limits, on compact subsets of $\mathbb{C}$, of polynomials with only real zeros.

Question:

1. Does it means that these function in $\mathcal{LP}$-class has only real zeroes?

2. Let $\phi(x)$ be an entire function but not in $\mathcal{LP}$-class. Does it means that $\phi(x)$ have a nonreal zero?

• Up to Wiki en.wikipedia.org/wiki/Laguerre-Pólya_class , the answer to Q1 is affirmative. – user64494 Aug 8 '17 at 4:41
• The answer to Q2 is no. One may take an entire function of infinite order with "many" real zeroes, say the Hadamard product $$\prod_{n=1}^\infty(1-x/\log(n+2))\exp(x/\log(n+2)+\cdots +1/n(x/\log(n+2))^n).$$The questions are not at the research level. – user64494 Aug 8 '17 at 6:33

To the second question the answer is no: $e^{z^2}$ and $ze^{z^2}$ are not in LP, but have only real zeros.
• Thanks for your answer! Here I have another question. If $\phi(x)$ be a real entire function of the form $e^{\alpha x^2}\phi_1(x)$, where $\alpha \geq 0$ and $\phi_1(x)$ has genus 0 or 1. If $\phi(x)$ does not belong to $\mathcal{LP}$, then $\phi(x)$ has a nonreal zero. Is it true? This comes from the proof of Theorem 2.2 in Iterated Laguerre and Turan inequalities (emis.de/journals/JIPAM/article191.html). – Aaron Jia Aug 10 '17 at 1:19
• The function you wrote never belongs to LP when $\alpha>0$. It belongs to LP if and only if $\alpha\leq 0$ and all zeros of $g$ are real. – Alexandre Eremenko Aug 10 '17 at 11:33