On a necessary condition for an entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre-Pólya class. (English) Zbl 1427.30048
Summary: For an entire function \(f(z) = \sum_{k = 0}^\infty a_k z^k\), \(a_k > 0\), we show that \(f\) does not belong to the Laguerre-Pólya class if the quotients \(\frac{a_{n - 1}^2}{a_{n - 2} a_n}\) are increasing in \(n\), and \(c := \underset{\lim}{n \rightarrow \infty} \frac{a_{n - 1}^2}{a_{n - 2} a_n}\) is smaller than an absolute constant \(q_\infty(q_\infty \approx 3 . 2336)\).
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) |
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