### A class of gap series with small growth in the unit disc.

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

Let p(z) be a polynomial of the form $p\left(z\right)={\sum}_{j=0}^{n}{a}_{j}{z}^{j}$, ${a}_{j}\in -1,1$. 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.

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...

We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball: For a function $u$ which is lower semi-continuous on $\partial {\mathbb{B}}^{n}$ we give necessary and sufficient conditions in order that there exists a holomorphic function $f\in \mathbb{O}\left({\mathbb{B}}^{n}\right)$ such that $$u\left(z\right)={\int}_{\left|\lambda \right|<1}{\left|f\left(\lambda z\right)\right|}^{2}\mathrm{d}{\U0001d50f}^{2}\left(\lambda \right).$$

We solve the following Dirichlet problem on the bounded balanced domain $\Omega $ with some additional properties: For $p>0$ and a positive lower semi-continuous function $u$ on $\partial \Omega $ with $u\left(z\right)=u\left(\lambda z\right)$ for $\left|\lambda \right|=1$, $z\in \partial \Omega $ we construct a holomorphic function $f\in \mathbb{O}\left(\Omega \right)$ such that $u\left(z\right)={\int}_{\mathbb{D}z}{\left|f\right|}^{p}d{\U0001d50f}_{\mathbb{D}z}^{2}$ for $z\in \partial \Omega $, where $\mathbb{D}=\{\lambda \in \u2102\phantom{\rule{0.222222em}{0ex}}|\lambda |<1\}$.

For $z\in \partial {B}^{n}$, the boundary of the unit ball in ${\u2102}^{n}$, let $\Lambda \left(z\right)=\left\{\lambda \phantom{\rule{0.222222em}{0ex}}\right|\lambda |\le 1\}$. If $f\in \mathbb{O}\left({B}^{n}\right)$ then we call $E\left(f\right)=\{z\in \partial {B}^{n}\phantom{\rule{0.222222em}{0ex}}{\int}_{\Lambda \left(z\right)}|f\left(z\right){|}^{2}\mathrm{d}\Lambda \left(z\right)=\infty \}$ the exceptional set for $f$. In this note we give a tool for describing such sets. Moreover we prove that if $E$ is a ${G}_{\delta}$ and ${F}_{\sigma}$ subset of the projective $(n-1)$-dimensional space ${\mathbb{P}}^{n-1}=\mathbb{P}\left({\u2102}^{n}\right)$ then there exists a holomorphic function $f$ in the unit ball ${B}^{n}$ so that $E\left(f\right)=E$.

We prove the following result which extends in a somewhat "linear" sense a theorem by Kierst and Szpilrajn and which holds on many "natural" spaces of holomorphic functions in the open unit disk 𝔻: There exist a dense linear manifold and a closed infinite-dimensional linear manifold of holomorphic functions in 𝔻 whose domain of holomorphy is 𝔻 except for the null function. The existence of a dense linear manifold of noncontinuable functions is also shown in any domain for its full space of holomorphic...

In this paper we give characterizations of those holomorphic functions in the unit disc in the complex plane that can be written as a quotient of functions in A(D), A∞(D) or Λ1(D) with a nonvanishing denominator in D. As a consequence we prove that if f ∈ Λ1(D) does not vanish in D, then there exists g ∈ Λ1(D) which has the same zero set as f in Dbar and such that fg ∈ A∞(D).

We consider the behavior of the power series ${}_{0}\left(z\right)={\sum}_{n=1}^{\infty}{\mu}^{2}\left(n\right){z}^{n}$ as z tends to $e\left(\beta \right)={e}^{2\pi i\beta}$ along a radius of the unit circle. If β is irrational with irrationality exponent 2 then ${}_{0}\left(e\left(\beta \right)r\right)=O\left({(1-r)}^{-1/2-\epsilon}\right)$. Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that ${}_{0}\left(e\left(\beta \right)r\right)=\Omega \left({(1-r)}^{-1+\delta}\right)$.

Consider the power series $\left(z\right)={\sum}_{n=1}^{\infty}\alpha \left(n\right)z\u207f$, where α(n) is a completely additive function satisfying the condition α(p) = o(lnp) for prime numbers p. Denote by e(l/q) the root of unity ${e}^{2\pi il/q}$. We give effective omega-estimates for $\left(e(l/{p}^{k})r\right)$ when r → 1-. From them we deduce that if such a series has non-singular points on the unit circle, then it is a zero function.

Let $\left(z\right)={\sum}_{n=1}^{\infty}\mu \left(n\right){z}^{n}$. We prove that for each root of unity $e\left(\beta \right)={e}^{2\pi i\beta}$ there is an a > 0 such that $\left(e\left(\beta \right)r\right)=\Omega \left({(1-r)}^{-a}\right)$ as r → 1-. For roots of unity e(l/q) with q ≤ 100 we prove that these omega-estimates are true with a = 1/2. From omega-estimates for (z) we obtain omega-estimates for some finite sums.

We prove, in a general framework, the existence of a closed infinite-dimensional subspace consisting of universal series.