Let ${{B}_{p1,p2}}\,=\,\left\{ z\,\in \,{{\mathbb{C}}^{n}}\,:\,{{\left| {{z}_{1}} \right|}^{{{p}_{1}}}}\,+\,{{\left| {{z}_{2}} \right|}^{{{p}_{2}}}}\,+\,\cdots \,+\,{{\left| {{z}_{n}} \right|}^{{{p}_{2}}}}\,<\,1 \right\}$ be an egg domain in ${{\mathbb{C}}^{n}}$. In this paper, we first characterize the Kobayashi metric on ${{B}_{{{p}_{1}},{{p}_{2}}}}\,\left( {{p}_{1}}\,\ge 1,\,{{p}_{2}}\,>\,1 \right)$ and then establish a new type of classical boundary Schwarz lemma at ${{z}_{0}}\in \partial {{B}_{{{p}_{1}},{{p}_{2}}}}$ for holomorphic self-mappings of ${{B}_{{{p}_{1}},{{p}_{2}}}}\,\left( {{p}_{1}}\,\ge \,1,\,{{p}_{2}}\,>\,1 \right)$), where ${{z}_{0}}={{\left( {{e}^{i\theta }},\,0,\ldots ,0 \right)}^{\prime }}$ and $\theta \,\in \,\mathbb{R}$.

Let φ and ψ be holomorphic self-maps of the unit polydisc Un in the n-dimensional complex space, and denote by Cφ and Cψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators Cφ−Cψ from Bloch space to bounded holomorphic function space in the unit polydisc. The compactness of the difference is also characterized.

We identify a class of domains of ${{\mathbb{C}}^{n}}$ in which any two commuting holomorphic self-maps have a common fixed point.

A function theoretic proof of Axler's zero multiplier theorem of Bergman spaces is given.

This paper proves that if р < s, then 0 is the only function that multiplies a Bergman Lр space into a Bergman Ls space.

We define a notion of measure for polynomials in several variables, basing our construction on the geometry of the zero-set of the polynomial. For polynomials in one variable, this measure reduces to the usual one. We begin the development of the theory of this measure along lines parallel to the theory of Mahler’s measure, indicating the differences and similarities between the two.

Let P be a polynomial in one variable with complex coefficients, P(z) = a0Π(z - αj). The measure of P, denoted M(P), is defined by

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As is well-known, an application of Jensen’s formula yields

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