On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre-P\'olya I class

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 $s a t i s f y t h e c o n d i t i o n$ q_2 \leq q_3, $t h e n$ f $h a s a t l e a s t o n e z e r o i n t h e s e g m e n t$ [-\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 $k = 2, 3, 4.$

On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre-P\'olya I class

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On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre-P\'olya I class

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On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre-P\'olya I class

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On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre-P\'olya I class

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