seeking reference on a theorem about sufficient conditions for an entire function with real coefficients to have only real zeros

Question

I am seeking reference(s) on the following theorem about sufficient conditions for an entire function with real coefficients to have only real zeros.

Theorem: Let $f_n(z)=\sum_0^n a_m z^m$ (with $a_m$ real) be the partial sums of the entire function $f(z)=\sum_0^\infty a_n z^n$:

If

(1) $f_n(z)$ have real zeros for $n\ge n_0>0$;

(2) the complex zeros $z_k=x_k+i y_k$ of $f_n(z)$ have the following property:

$$|y_k|\ge c n^a,\text{ } (a,c>0)$$

Then all the zeros of $f(z)$ are real.

Because when $n\to \infty$, we have $$|z_k|^2=x_k^2+y_k^2 \ge c^2 n^{2a} \to \infty$$ thus all the complex zeros are pushed up to infinity and the zeros survived are all real.

The entire functions $\sin z$ and $\cos z$ seem to fall into this category.

Answer 1

I don't have a reference to that one specifically, but much more generally, this is a corollary of Hurwitz's theorem (e.g. Conway, "Functions of One Complex Variable", VII.2.5-6): if $f$ and $f_n$ are analytic functions on a domain $D$, $f_n \to f$ uniformly on compacta, and all but finitely many $f_n$ have no zeros in $D$, then either $f$ is identically zero or $f$ has no zeros in $D$.