An Estimate for Minkowski Balances of Homogeneous Polynomials and an Application

Abstract The paper refers to an aggregated estimate of two initial terms in the power series expansions of holomorphic functions and mappings of several complex variables. The authors solve first this problem in some Bavrin’s families of functions, i.e., functions $$f:\mathcal {G}\rightarrow \mathbb {C},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>G</mml:mi> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> holomorphic in bounded complete n -circular domains $$\mathcal {G}\subset \mathbb {C}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> and satisfying conditions, similar as in geometric function theory of one variable. Next, they apply this result to a family of mappings $$F:\mathbb {B} ^{n}\rightarrow $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>→</mml:mo> </mml:mrow> </mml:math> $$\mathbb {C}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> biholomorphic in the open unit Euclidean ball $$\mathbb {B}^{n}\subset \mathbb {C}^{n}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>