Multidimensional analogues of the refined Bohr type inequalities

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk $\mathbb{D}^n$.We also prove two other sharp versions of the Bohr inequality in the setting of several complex variables: one by replacing the constant term with the absolute value of the function, and another by replacing it with the square of the absolute value of the function.Furthermore, we establish multidimensional analogues of known results concerning the modulus of the derivative of analytic functions in the unit disk $\mathbb{D}$, replacing the derivative with the radial derivative of holomorphic functions in $\mathbb{D}^n$.All of the established results are shown to be sharp.

💡 Research Summary

The paper investigates refined versions of the classical Bohr inequality in the setting of several complex variables, focusing on holomorphic functions defined on the unit polydisk ( \mathbb{D}^n ). After a concise review of Bohr’s original theorem, its one‑dimensional refinements, and earlier multidimensional extensions (notably the Boas‑Khavinson n‑dimensional Bohr radius), the authors set up the necessary notation for multi‑indices, the supremum norm ( |z|\infty ), and the Euler (radial) derivative ( Df(z)=\sum{k=1}^{n}z_k\partial f/\partial z_k ).

The main contributions are four families of theorems that generalize recent refined Bohr‑type results (Theorems B, C, E, F in the literature) to several variables.

1. Theorem 2.1 provides a multidimensional analogue of the refined Bohr‑Rogosinski inequality. For a bounded holomorphic map ( f(z)=\sum_{|\alpha|\ge0}a_\alpha z^\alpha ) with ( |f(z)|\le1 ) on the closed polydisk, the authors define two quantities ( A_1(z,r) ) and ( A_2(z,r) ) that combine the absolute value of ( f ), the sums of absolute coefficients of degree ( \ge N ), and weighted squares of lower‑degree coefficients. The admissible radius ( R_{n,N} ) (resp. ( R’{n,N} )) is the positive root of the equation ( \psi_N(nr)=0 ) (resp. ( (1+nr)(nr)^N-(1-nr)^2=0 )), where ( r=|z|\infty ). The authors prove that these radii are optimal by constructing extremal functions that attain equality.

2. Theorem 2.2 replaces the constant term ( |a_0| ) in the classical Bohr sum by the full modulus ( |f(z)| ) or its square ( |f(z)|^2 ). The resulting inequalities ( A_3 ) and ( A_4 ) hold for ( r\le n r_{a_0} ) and ( r\le n r’{a_0} ) respectively, where ( n r{a_0}=2/3+|a_0|+\sqrt{5}(1+|a_0|) ) and ( n r’{a_0} ) is the unique positive root of a cubic equation involving ( |a_0| ). The authors show that ( n r{a_0}>\sqrt{5}-2 ), improving the one‑dimensional bound.

3. Theorem 2.3 incorporates the radial derivative ( Df ). The quantities ( I(z,r) ) and ( J(z,r) ) contain ( |f(z)| ), ( |Df(z)| ), higher‑order coefficient sums, and a correction term involving the squares of coefficients. The admissible radius is the universal constant ( \sqrt{17}-3/4 ) for ( I ), exactly as in the one‑dimensional refined result, while for ( J ) the radius ( n r_0 ) is the positive root of ( 1-2nr-(nr)^2-(nr)^3-(nr)^4=0 ). Both constants are shown to be best possible.

4. Theorem 2.4 deals with higher‑order mixed partial derivatives. For any integer ( N\ge2 ) the authors define ( M(z,r) ) and ( N(z,r) ) that involve the sums of absolute values of the normalized derivatives ( \partial^{|\alpha|}f/\partial z^\alpha ) multiplied by the monomials ( |z|^\alpha ). The admissible radii ( \tilde R_{n,N} ) and ( \tilde R’_{n,N} ) are the smallest positive solutions of the equations
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