A characterization is given of the sequences
{
γ
k
}
k
=
0
∞
\{ {\gamma _k}\}_{k = 0}^\infty
with the property that, for any complex polynomial
f
(
z
)
=
Σ
a
k
z
k
f(z) = \Sigma \,{a_k}{z^k}
and convex region
K
K
containing the origin and the zeros of
f
f
, the zeros of
Σ
γ
k
a
k
z
k
\Sigma \,{\gamma _k}{a_k}{z^k}
again lie in
K
K
. Many applications and related results are also given. This work leads to a study of the Taylor coefficients of entire functions of type
I
\text {I}
in the Laguerre-Pólya class. If the power series of such a function is given by
Σ
γ
k
z
k
/
k
!
\Sigma \,{\gamma _k}{z^k}/k!
and the sequence
{
γ
k
}
\{ {\gamma _k}\}
is positive and increasing, then the sequence satisfies an infinite collection of strong conditions on the differences, namely
Δ
n
γ
k
⩾
0
{\Delta ^n}{\gamma _k} \geqslant 0
for all
n
n
,
k
k
.