On entire functions having Taylor sections with only real zeros. (English) Zbl 1078.30022
Let \(S^\ast\) be the set of all entire \(f(z)=\sum_{k=0}^\infty a_kz^k\) such that \(a_k>0\) and that an unbounded sequence \(\{n_k\}\) exists such that the \(n_k\) partial sums \(S_{n_k}\) have only real zeros. For a given \(f\) as above, define the sequence
\[
q_n(f) := \frac{a_{n-1}^2}{a_{n-2}a_n}
\]
for \(n\geq 2\). The main result of the article states that if \(f\in S^\ast\) and if the limit \(q_0:=\lim_{n\to\infty} q_n(f)\) exists, then for every positive integer \(m\) we have that the polynomial
\[
\sum_{k=0}^m \frac{z^k}{k!(\sqrt{q_0})^{k^2}}
\]
only has real roots. As a corollary of this and other observations, the authors prove that if \(f\in S^\ast\) and if the limit \(q_0:=\lim_{n\to\infty} q_n(f)\) exists, then \(q_0\geq q_\infty\), where \(q_\infty \approx 3.23\).
Reviewer: Vania Mascioni (Muncie)
MSC:
30D15 | Special classes of entire functions of one complex variable and growth estimates |
30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
26C10 | Real polynomials: location of zeros |