The purpose of this note is to obtain an improved lower bound for the multidimensional Bohr radius introduced by L. Aizenberg (2000, Proceedings of the American Mathematical Society 128, 1147–1155), by means of a rather simple argument.

In this article, we study a generalized Bohr radius $R_{p, q}(X), p, q\in [1, \infty )$ defined for a complex Banach space X. In particular, we determine the exact value of $R_{p, q}(\mathbb {C})$ for the cases (i) $p, q\in [1, 2]$ , (ii) $p\in (2, \infty ), q\in [1, 2]$ , and (iii) $p, q\in [2, \infty )$ . Moreover, we consider an n-variable version $R_{p, q}^n(X)$ of the quantity $R_{p, q}(X)$ and determine (i) $R_{p, q}^n(\mathcal {H})$ for an infinite-dimensional complex Hilbert space $\mathcal {H}$ and (ii) the precise asymptotic value of $R_{p, q}^n(X)$ as $n\to \infty $ for finite-dimensional X. We also study the multidimensional analog of a related concept called the p-Bohr radius. To be specific, we obtain the asymptotic value of the n-dimensional p-Bohr radius for bounded complex-valued functions, and in the vector-valued case, we provide a lower estimate for the same, which is independent of n.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle ${\mathbb T}$ , there is a function f holomorphic in the unit disc ${{\mathbb D}}$ , having $\psi $ as radial limit a.e. on ${\mathbb T}$ . We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point p is replaced by (asymptotically) total approach to p.

We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then $f=g$ up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi $ . We also prove that if f and g are functions in the Nevanlinna class, and if $|f|=|g|$ on the unit circle and on a circle inside the unit disc, then $f=g$ up to the multiplication of a unimodular constant.

Let $\unicode[STIX]{x1D6FA}$ be a member of a certain class of convex ellipsoids of finite/infinite type in $\mathbb{C}^{2}$. In this paper, we prove that every holomorphic function in $L^{p}(\unicode[STIX]{x1D6FA})$ can be approximated by holomorphic functions on $\bar{\unicode[STIX]{x1D6FA}}$ in $L^{p}(\unicode[STIX]{x1D6FA})$-norm, for $1\leq p<\infty$. For the case $p=\infty$, the continuity up to the boundary is additionally required. The proof is based on $L^{p}$ bounds in the additive Cousin problem.

We study function multipliers between spaces of holomorphic functions on the unit disc of the complex plane generated by symmetric sequence spaces. In the case of sequence $\ell ^{p}$ spaces we recover Nikol’skii’s results [‘Spaces and algebras of Toeplitz matrices operating on $\ell ^{p}$’, Sibirsk. Mat. Zh.7 (1966), 146–158].

A natural topology on the set of germs of holomorphic functions on a compact subset $K$ of a Fréchet space is the locally convex inductive limit topology of the spaces $\mathcal{O}(\sOm)$ endowed with the compact open topology; here $\sOm$ is any open subset containing $K$. Mujica gave a description of this space as the inductive limit of a suitable sequence of compact subsets. He used a set of intricate semi-norms for this. We give a projective characterization of this space, using simpler semi-norms, whose form is similar to the one used in the Whitney Extension Theorem for $C_\infty$ functions. They are quite natural in a framework where extensions are involved. We also give a simple proof that this topology is strictly stronger than the topology of the projective limit of the non-quasi-analytic spaces.

AMS 2000 Mathematics subject classification: Primary 46A13; 46F15

We give a characterization of composition operators between algebras of analytic functions on a Banach space. We show (under fairly general conditions) that they are precisely the multiplicative operators that are transposes of operators of between the preduals of the algebras. The special cases of $H^\infty(U)$ and $H_{\mathrm{b}}(U)$ are considered. In these cases, the composition operators are those which are pointwise-to-pointwise continuous and/or $\tau_0$-to-$\tau_0$ continuous (where $\tau_0$ is the compact-open topology). We obtain Banach–Stone-type theorems for these algebras.

It is shown that if E, F are Fréchet spaces, E ∈ (Hub), F ∈ (DN) then H(E, F) = Hub(E, F) holds. Using this result we prove that a Fréchet space E is nuclear and has the property (Hub) if and only if every entire function on E with values in a Fréchet space F ∈ (DN) can be represented in the exponential form. Moreover, it is also shown that if H(F*) has a LAERS and E ∈ (Hub) then H(E × F*) has a LAERS, where E, F are nuclear Fréchet spaces, F* has an absolute basis, and conversely, if H(E × F*) has a LAERS and F ∈ (DN) then E ∈ (Hub).