Abstract:
Let
$K\subset \mathbb{C}$
be a polynomially convex compact set,
$f$
be a
function analytic in a domain
$\overline{\mathbb{C}}\setminus K$
with
Taylor expansion
$f(z) =\sum_{k=0}^{\infty }a_{k}/z^{k+1}
$ at
$\infty
$, and
$H_{i}(f) :=\det (a_{k+l})
_{k,l=0}^{i}$
be the related Hankel determinants.
The classical Polya theorem
[11] says that
$$
\limsup_{i\to \infty }\vert H_{i}(f) \vert
^{1/i^{2}}\leq d(K) ,
$$
where
$d(K)
$ is the transfinite diameter of
$K$.
The main result of this paper is
a multivariate analog of the Polya inequality for a weighted Hankel-type determinant
constructed from the Taylor series of a function analytic on a
$\mathbb{C}$-convex (=
strictly linearly convex) domain in
$\mathbb{C}^{n}$.
