Transfinite diameters and polya inequality
Günyüz, Ozan (2015) Transfinite diameters and polya inequality. [Thesis]
This dissertation deals with two main problems concerning Polya's inequality, mostly, in several variables. We investigate the problems about obtaining the new version of Polya inequality for domains in terms of internal transfinite diameter, due to V. Zakharyuta, and the sharpness of Polya inequality in one and multivariable case. First part is devoted to the sharpness of Polya's inequality. We make a classiffication of sharpness properties of a Polya's inequality related to a compact set in multivariate case and examine the stability of these properties by using the considerations obtained from the stability of trans nite diameter with respect to the approximations from inside and outside by compact sets. For real compact sets in Cn, we prove that they have the strong sharpness property. The main ingredient we exploit in proving this is the Bloom-Levenberg integral representation of Vandermondians. In the second part of thesis, we study internal characteristics of domains in Cn: As a consequence of classical Polya's inequality, we give first the new version of Polya inequality including the internal transfinite diameter in one variable. For multivariable case, given a linearly convex domain with an approximation of sufficiently good sets from inside, it is proved that the internal transfinite diameter of boundary viewed from a point is equal to the transfinite diameter of the compact conjugate set to the aforementioned domain. This will enable us to establish the domain analogue of Polya inequality involving internal transfinite diameter for domains called linearly convex by using the duality due to Aizenberg-Martineau.
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