The Internal Polya Inequality for $\mathbb{C}$-convex Domains in $\mathbb{C}^n$
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2016
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910299986067456 |
|---|---|
| author | Günyüz, Ozan Zakharyuta, Vyacheslav |
| author_facet | Günyüz, Ozan Zakharyuta, Vyacheslav |
| contents | Let $K\subset \mathbb{C}$ be a polynomially convex compact set, $f$ be a function analytic in a domain $\overline{\mathbb{C}}\smallsetminus K$ with Taylor expansion $f\left( z\right) =\sum_{k=0}^{\infty }\frac{a_{k}}{z^{k+1}} $ at $\infty $, and $H_{s}\left( f\right) :=\det \left( a_{k+l}\right) _{k,l=0}^{s}$ related Hankel determinants. The classical Polya theorem \cite% {P} says that \[ \limsup_{s\rightarrow \infty }\left\vert H_{s}\left( f\right) \right\vert ^{1/s^{2}}\leq d\left( K\right) , \]% where $d\left( K\right) $ is the transfinite diameter of $K$. The main result of this paper is multivariate internal analogs of Polya's inequality for $\mathbb{C}$-convex (=strictly linearly convex) domains $D\subset \mathbb{C}^{n}$ and weighted Hankel-type determinants, constructed from the Taylor coefficients of a function $f\in A\left( D\right) $ at a given point $% a\in D$; therewith the weights are generated by $s$-indicatrices of the sequence of analytic functionals biorthogonal to the system of monomials in $% \mathbb{C}^{n}$. It is proved by the reduction to the outer multivariate analog of Polya's inequality (Zakharyuta, Math. USSR Sbornik, \textbf{25 }% (1975)) and is based on the characterization of the strict linear convexity in terms of $s$-indicatrices (S. Znamenskii, Siberian Math. J. \textbf{26 } (1985)). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1606_00186 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | The Internal Polya Inequality for $\mathbb{C}$-convex Domains in $\mathbb{C}^n$ Günyüz, Ozan Zakharyuta, Vyacheslav Complex Variables Let $K\subset \mathbb{C}$ be a polynomially convex compact set, $f$ be a function analytic in a domain $\overline{\mathbb{C}}\smallsetminus K$ with Taylor expansion $f\left( z\right) =\sum_{k=0}^{\infty }\frac{a_{k}}{z^{k+1}} $ at $\infty $, and $H_{s}\left( f\right) :=\det \left( a_{k+l}\right) _{k,l=0}^{s}$ related Hankel determinants. The classical Polya theorem \cite% {P} says that \[ \limsup_{s\rightarrow \infty }\left\vert H_{s}\left( f\right) \right\vert ^{1/s^{2}}\leq d\left( K\right) , \]% where $d\left( K\right) $ is the transfinite diameter of $K$. The main result of this paper is multivariate internal analogs of Polya's inequality for $\mathbb{C}$-convex (=strictly linearly convex) domains $D\subset \mathbb{C}^{n}$ and weighted Hankel-type determinants, constructed from the Taylor coefficients of a function $f\in A\left( D\right) $ at a given point $% a\in D$; therewith the weights are generated by $s$-indicatrices of the sequence of analytic functionals biorthogonal to the system of monomials in $% \mathbb{C}^{n}$. It is proved by the reduction to the outer multivariate analog of Polya's inequality (Zakharyuta, Math. USSR Sbornik, \textbf{25 }% (1975)) and is based on the characterization of the strict linear convexity in terms of $s$-indicatrices (S. Znamenskii, Siberian Math. J. \textbf{26 } (1985)). |
| title | The Internal Polya Inequality for $\mathbb{C}$-convex Domains in $\mathbb{C}^n$ |
| topic | Complex Variables |
| url | https://arxiv.org/abs/1606.00186 |