turkmath.org

Suppose $f$ is a function analytic in a domain $\overline{\mathbb{C}} \smallsetminus K$, where $K\subset \mathbb{C}$ is a polynomially convex compact set, with Taylor expansion $f\left( z\right) =\sum_{k=0}^{\infty } \frac{a_{k}}{z^{k+1}}$ at $\infty $, and $H_{s}\left( f\right) :=\det \left( a_{k+l}\right) _{k,l=0}^{s}$ are related Hankel determinants. The classical Polya theorem (1929) says that \[ D\left( f\right) :=\limsup_{s\rightarrow \infty }\left\vert H_{s}\left( f\right) \right\vert ^{1/s^{2}}\leq d\left( K\right) , \] where $d\left( K\right) $ is the transfinite diameter of $K$ introduced by Fekete in 1923. In our talk will be considered some multivariate internal analogs of this result. They are based on the generalized Polya inequality for compacta and on the notion of internal transfinite diameter $d\left( a,\partial D\right) $ of the boundary of a domain $D\subset \mathbb{C}^{n}$ viewed from a given point $a\in D$. The presented results are joint with Ozan Günyüz.