Author:
Nguyen Thu Hien,Vishnyakova Anna
Abstract
AbstractWe find intervals $$[\alpha , \beta (\alpha )]$$
[
α
,
β
(
α
)
]
such that, if a univariate real polynomial or entire function $$f(z) = a_0 + a_1 z + a_2 z^2 + \cdots $$
f
(
z
)
=
a
0
+
a
1
z
+
a
2
z
2
+
⋯
with positive coefficients satisfies the conditions $$ q_k(f) = {a_{k-1}^2}/({a_{k-2}a_{k}}) \in [\alpha , \beta (\alpha )]$$
q
k
(
f
)
=
a
k
-
1
2
/
(
a
k
-
2
a
k
)
∈
[
α
,
β
(
α
)
]
for all $$k \geqslant 2$$
k
⩾
2
, then f belongs to the Laguerre–Pólya class. For instance, from Hutchinson’s theorem, one can observe that f belongs to the Laguerre–Pólya class (has only real zeros) when $$q_k(f) \in [4, + \infty )$$
q
k
(
f
)
∈
[
4
,
+
∞
)
. We are interested in finding those intervals which are not subsets of $$[4, + \infty )$$
[
4
,
+
∞
)
.
Funder
Mathematisches Forschungsinstitut Oberwolfach
Julius-Maximilians-Universität Würzburg
Publisher
Springer Science and Business Media LLC
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