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Cauchy-Poisson transform and polynomial inequalities

Mirosław Baran (2009)

Annales Polonici Mathematici

We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in N is Hölder continuous then E admits a Szegö type inequality with weight function d i s t ( x , E ) - ( 1 - κ ) with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

Chebyshev and Robin constants on algebraic curves

Jesse Hart, Sione Ma`u (2015)

Annales Polonici Mathematici

We define directional Robin constants associated to a compact subset of an algebraic curve. We show that these constants satisfy an upper envelope formula given by polynomials. We use this formula to relate the directional Robin constants of the set to its directional Chebyshev constants. These constants can be used to characterize algebraic curves on which the Siciak-Zaharjuta extremal function is harmonic.

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