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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?

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    $\begingroup$ Up to Wiki en.wikipedia.org/wiki/Laguerre-Pólya_class , the answer to Q1 is affirmative. $\endgroup$ – user64494 Aug 8 '17 at 4:41
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    $\begingroup$ 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. $\endgroup$ – user64494 Aug 8 '17 at 6:33
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To the first question the answer is yes: the limit of functions with all zeros on a closed set has zeros on the same closed set, if this limit is not identically zero.

To the second question the answer is no: $e^{z^2}$ and $ze^{z^2}$ are not in LP, but have only real zeros.

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  • $\begingroup$ 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). $\endgroup$ – Aaron Jia Aug 10 '17 at 1:19
  • $\begingroup$ 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. $\endgroup$ – Alexandre Eremenko Aug 10 '17 at 11:33

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