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?



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


