Thu Hien Nguyen, Anna Vishnyakova
Published 2020-01-16Version 1
For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we show that if $f$ belongs to the Laguerre-P\'olya class, and the quotients $q_k := \frac{a_{k-1}^2}{a_{k-2}a_k}, k=2, 3, \ldots $ satisfy the condition $q_2 \leq q_3,$ then $f$ has at least one zero in the segment $[-\frac{a_1}{a_2},0].$ We also give necessary conditions and sufficient conditions of the existence of such a zero in terms of the quotients $q_k$ for $k=2,3, 4.$
Comments: 18 pages. Submitted to Results in Mathematics. Partially supported by the Akhiezer Foundation. arXiv admin note: substantial text overlap with arXiv:1903.09070, arXiv:1912.10035
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