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A Dichotomy Principle for Universal Series

V. Farmaki, V. Nestoridis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ( α j ) j = 1 of scalars, there exists a subsequence ( α k j ) j = 1 such that either every subsequence of ( α k j ) j = 1 defines a universal series, or no subsequence of ( α k j ) j = 1 defines a universal series. In particular examples we decide which of the two cases holds.

A note on the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

Let p(z) be a polynomial of the form p ( z ) = j = 0 n a j z j , a j - 1 , 1 . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

Approximation par des fonctions holomorphes à croissance contrôlée.

Philippe Charpentier, Yves Dupain, Modi Mounkaila (1994)

Publicacions Matemàtiques

Let Ω be a bounded pseudo-convex domain in Cn with a C∞ boundary, and let S be the set of strictly pseudo-convex points of ∂Ω. In this paper, we study the asymptotic behaviour of holomorphic functions along normals arising from points of S. We extend results obtained by M. Ortel and W. Schneider in the unit disc and those of A. Iordan and Y. Dupain in the unit ball of Cn. We establish the existence of holomorphic functions of given growth having a "prescribed behaviour" in almost all normals arising...

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