<p>In this article, we give bounds of the partial derivatives at the origin of pluriharmonic functions defined in the unit ball of the Minkowski space to the Euclidean unit ball in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {C}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. Next, we derive Schwarz-Pick estimates of arbitrary order partial derivatives for pluriharmonic functions from the Euclidean unit ball in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {C}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> or from the unit polydisk in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {C}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> into the Euclidean unit ball in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {C}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. Moreover, we establish some sharp coefficient type Schwarz-Pick estimates of arbitrary order partial derivatives for pluriharmonic functions with real part strictly less than 1 in the unit polydisk in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {C}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. Furthermore, we present a new bound of Schwarz-Pick estimates of arbitrary order partial derivatives for pluriharmonic functions in the Euclidean unit ball in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {C}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> to the unit ball of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>. Additionally, we find the exact asymptotic value of the mixed Bohr radii and the arithmetic Bohr radii of pluriharmonic functions. Finally, we obtain Rogosinski radius for harmonic or pluriharmonic functions.</p>

Schwarz-Pick Estimates for Pluriharmonic Functions and Their Consequences

期刊论文
发表日期：

2026年1月30日

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Shankey Kumar,
Saminathan Ponnusamy

摘要

In this article, we give bounds of the partial derivatives at the origin of pluriharmonic functions defined in the unit ball of the Minkowski space to the Euclidean unit ball in $\mathbb {C}^N$ $C^{N}$ . Next, we derive Schwarz-Pick estimates of arbitrary order partial derivatives for pluriharmonic functions from the Euclidean unit ball in $\mathbb {C}^N$ $C^{N}$ or from the unit polydisk in $\mathbb {C}^N$ $C^{N}$ into the Euclidean unit ball in $\mathbb {C}^N$ $C^{N}$ . Moreover, we establish some sharp coefficient type Schwarz-Pick estimates of arbitrary order partial derivatives for pluriharmonic functions with real part strictly less than 1 in the unit polydisk in $\mathbb {C}^N$ $C^{N}$ . Furthermore, we present a new bound of Schwarz-Pick estimates of arbitrary order partial derivatives for pluriharmonic functions in the Euclidean unit ball in $\mathbb {C}^N$ $C^{N}$ to the unit ball of $\mathbb {C}$ $C$ . Additionally, we find the exact asymptotic value of the mixed Bohr radii and the arithmetic Bohr radii of pluriharmonic functions. Finally, we obtain Rogosinski radius for harmonic or pluriharmonic functions.