A classical result of Hardy and Littlewood states that if $f\left(z\right)={\sum }_{m=0}^{\infty }{a}_m{z}^m$ is in $H^p$, 0 < p ≤ 2, of the unit disk of ℂ, then ${\sum }_{m=0}^{\infty }{(m+1)}^{p-2}{\left|a_m\right|}^p\le c_p\parallel f{\parallel }_p^p$ where $c_p$ is a positive constant depending only on p. In this paper, we provide an extension of this result to Hardy and weighted Bergman spaces in the unit ball of ${ℂ}^n$, and use this extension to study some related multiplier problems in ${ℂ}^n$.

A characterization of a generalized order of analytic functions of several complex variables by means of polynomial approximation and interpolation is established.

We discuss an example of an open subset of a torus which admits a dense entire curve, but no dense Brody curve.

Let K be a compact set in ℂ, f a function analytic in ℂ̅∖K vanishing at ∞. Let $f\left(z\right)={\sum }_{k=0}^{\infty }{a}_k{z}^{-k-1}$ be its Taylor expansion at ∞, and $H_s\left(f\right)=det{\left(a_{k+l}\right)}_{k,l=0}^s$ the sequence of Hankel determinants. The classical Pólya inequality says that $limsu{p}_{s\to \infty }{\left|H_s\left(f\right)\right|}^{1/s²}\le d\left(K\right)$, where d(K) is the transfinite diameter of K. Goluzin has shown that for some class of compacta this inequality is sharp. We provide here a sharpness result for the multivariate analog of Pólya’s inequality, considered by the second author in Math. USSR Sbornik 25 (1975), 350-364.

By considering a question proposed by F. Gross concerning unique range sets of entire functions in $ℂ$, we study the unicity of meromorphic functions in ${ℂ}^m$ that share three distinct finite sets CM and obtain some results which reduce $5\le c_3\left(ℳ\left({ℂ}^m\right)\right)\le 9$ to $5\le c_3\left(ℳ\left({ℂ}^m\right)\right)\le 6$.

The necessary and sufficient condition that a given plurisubharmonic or a subharmonic function admits the representation by the logarithmic potential (up to pluriharmonic or a harmonic term) is obtained in terms of the Radon transform. This representation is applied to the problem of representation of analytic functions by products of primary factors.

In this paper, we study the behaviour near the boundary of the complex tangent coefficients of a closed positive current in a bounded domain of C3 with C∞ boundary. Assuming that the current satisfies the Blaschke condition, we give a condition on the complex tangent coefficients which is better than the one which can be proved using the pseudo-distance introduced by A. Nagel, E. Stein and S. Wainger (in analogy with the case of domains in C2). Moreover, when the domain is supposed to be pseudoconvex,...

We prove some theorems on uniqueness of meromorphic mappings into complex projective space ℙⁿ(ℂ), which share 2n+3 or 2n+2 hyperplanes with truncated multiplicities.