For an entire function $$f(z) = \sum _{k=0}^\infty a_k z^k, a_k>0$$ f ( z ) = ∑ k = 0 ∞ a k z k , a k > 0 , we show that if f belongs to the Laguerre–Pólya class, and the quotients $$q_k:= \frac{a_{k-1}^2}{a_{k-2}a_k}, k=2, 3, \ldots $$ q k : = a k - 1 2 a k - 2 a k , k = 2 , 3 , … satisfy the condition $$q_2 \le q_3$$ q 2 ≤ q 3 , then f has at least one zero in the segment $$[-\frac{a_1}{a_2},0]$$ [ - 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 $$q_k$$ q k for $$k=2,3, 4$$ k = 2 , 3 , 4 .

中文翻译：

---

关于 Laguerre-Pólya I 类全函数的泰勒系数的最接近零根和二商

对于整个函数 $$f(z) = \sum _{k=0}^\infty a_k z^k, a_k>0$$ f ( z ) = ∑ k = 0 ∞ akzk , ak > 0 ，我们证明如果 f 属于 Laguerre–Pólya 类，并且商 $$q_k:= \frac{a_{k-1}^2}{a_{k-2}a_k}, k=2, 3, \ldots $ $ qk := ak - 1 2 ak - 2 ak , k = 2 , 3 , ... 满足条件 $$q_2 \le q_3$$ q 2 ≤ q 3 ，则 f 在 $$[ 段中至少有一个零-\frac{a_1}{a_2},0]$$ [ - a 1 a 2 , 0 ] 。对于 $$k=2,3, 4$$ k = 2 , 3 , 4 ，我们还根据商 $$q_k$$ qk 给出了这种零存在的必要条件和充分条件。