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On the maximum modulus theorem for nonanalytic functions in several complex variables.

Mario O. González Rodríguez (1981)

Revista Matemática Hispanoamericana

Let w = f(z1, ..., zn) = u(x1, ..., yn) + iv(x1, ..., yn) be a complex function of the n complex variables z1, ..., zn, defined in some open set A ⊂ Cn. The purpose of this note is to prove a maximum modulus theorem for a class of these functions, assuming neither the continuity of the first partial derivatives of u and v with respect to xk and yk, nor the conditions fzk = 0 in A for k = 1, 2, ..., n (the Cauchy-Riemann equations in complex form).

On the Rogosinski radius for holomorphic mappings and some of its applications

Lev Aizenberg, Mark Elin, David Shoikhet (2005)

Studia Mathematica

The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than 1 in the open unit disk: | ∑ n = 0 ∞ a ₙ z ⁿ | < 1 , |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: | ∑ n = 0 k a ₙ z ⁿ | < 1 , |z| < 1/2, and this radius is sharp. We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski’s theorem as well as some applications to dynamical systems are considered....

On the spectrum of A(Ω) and H ∞ ( Ω )

Urban Cegrell (1993)

Annales Polonici Mathematici

We study some properties of the maximal ideal space of the bounded holomorphic functions in several variables. Two examples of bounded balanced domains are introduced, both having non-trivial maximal ideals.

On the Toëplitz corona problem.

Eric Amar (2003)

Publicacions Matemàtiques

The aim of this note is to characterize the vectors g = (g1, . . . ,gk) of bounded holomorphic functions in the unit ball or in the unit polydisk of Cn such that the Corona is true for them in terms of the H2 Corona for measures on the boundary.

On the zero set of the Kobayashi-Royden pseudometric of the spectral unit ball

Nikolai Nikolov, Pascal J. Thomas (2008)

Annales Polonici Mathematici

Given A∈ Ωₙ, the n²-dimensional spectral unit ball, we show that if B is an n×n complex matrix, then B is a “generalized” tangent vector at A to an entire curve in Ωₙ if and only if B is in the tangent cone C A to the isospectral variety at A. In the case of Ω₃, the zero set of the Kobayashi-Royden pseudometric is completely described.

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