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arxiv: 2308.15913 · v4 · submitted 2023-08-30 · 🧮 math.CV

The Landau-Bloch type theorems for certain class of holomorphic and pluriharmonic mappings in mathbb{c}^n

Vasudeva Rao Allu , Rohit Kumar This is my paper

Pith reviewed 2026-05-24 07:24 UTC · model grok-4.3

classification 🧮 math.CV
keywords Bloch constantLandau-Bloch theoremholomorphic mappingspluriharmonic mappingsunit ballseveral complex variablesunivalent functions
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The pith

Lower estimates for Bloch's constant are derived for two new classes of holomorphic mappings on the unit ball in C^n together with Landau-Bloch theorems for pluriharmonic subclasses.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines two classes of holomorphic mappings on the unit ball B^n of n-dimensional complex space. It obtains lower estimates for Bloch's constant associated to these classes. Landau-Bloch type theorems are also derived for some subclasses of pluriharmonic mappings on B^n. A sympathetic reader would care because these results provide explicit quantitative control on how large the images of such mappings can be while remaining univalent or bounded in higher dimensions.

Core claim

We define two classes of holomorphic mappings on the unit ball B^n and obtain the lower estimates for Bloch's constant for these classes. We also derive the Landau-Bloch type theorem for some subclasses of pluriharmonic mappings defined on the unit ball B^n.

What carries the argument

The two classes of holomorphic mappings on B^n for which lower estimates of Bloch's constant are proved, and the subclasses of pluriharmonic mappings for which Landau-Bloch theorems are established.

If this is right

  • The Bloch constant for the defined holomorphic classes is bounded from below by a positive constant depending on n.
  • Landau-Bloch theorems give bounds on the size of the domain where the pluriharmonic mappings are univalent or satisfy certain growth conditions.
  • The results hold for mappings in the specified classes on the unit ball in C^n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the lower estimates are sharp, they could provide the exact Bloch constant for these classes.
  • The approach may extend to other classes defined by different growth conditions in several complex variables.
  • Testing the theorems numerically for low dimensions like n=2 could reveal if the bounds are optimal.

Load-bearing premise

The definitions chosen for the two classes of holomorphic mappings ensure that the lower estimates for Bloch's constant and the Landau-Bloch theorems are valid.

What would settle it

Constructing or identifying a mapping in one of the two holomorphic classes whose Bloch constant is strictly less than the paper's lower estimate would falsify the result.

read the original abstract

In this paper, we first define two classes of holomorphic mappings defined on the unit ball $B^n$ of n-dimensional complex space $\mathbb{C}^n$ and obtain the lower estimates for Bloch's constant for these classes. Also, we derive the Landau-Bloch type theorem for some subclasses of pluriharmonic mappings defined on the unit ball $B^n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript defines two new classes of holomorphic mappings on the unit ball B^n in C^n and derives lower estimates for the associated Bloch constants. It also establishes Landau-Bloch type theorems for certain subclasses of pluriharmonic mappings on B^n.

Significance. If the newly defined classes are non-empty and the membership conditions permit non-trivial applications of Bloch-radius arguments, the lower estimates would extend classical one-variable results to several complex variables and could be of interest to researchers working on geometric function theory in higher dimensions. The Landau-Bloch theorems for pluriharmonic subclasses would similarly add to the literature if the subclasses are meaningfully restricted.

major comments (1)
  1. [§2 (definitions of the classes)] The central claims rest on the precise coefficient or growth conditions used to define the two holomorphic classes (presumably introduced in §2). The abstract and available description provide no explicit statement of these conditions (e.g., bounds on ||f'(0)||, Jacobian norms, or subordination relations), making it impossible to verify that the classes are non-empty, that the lower estimates follow from the definitions without circularity, or that the estimates do not reduce to trivial or previously known cases by construction.
minor comments (1)
  1. Notation for the unit ball B^n and the pluriharmonic subclasses should be introduced with explicit references to prior literature on Bloch constants in several variables to clarify novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need for greater clarity regarding the definitions of the new classes. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§2 (definitions of the classes)] The central claims rest on the precise coefficient or growth conditions used to define the two holomorphic classes (presumably introduced in §2). The abstract and available description provide no explicit statement of these conditions (e.g., bounds on ||f'(0)||, Jacobian norms, or subordination relations), making it impossible to verify that the classes are non-empty, that the lower estimates follow from the definitions without circularity, or that the estimates do not reduce to trivial or previously known cases by construction.

    Authors: The two classes of holomorphic mappings are defined explicitly in Section 2 of the manuscript via specific coefficient bounds on the Taylor expansions and growth restrictions on the derivatives (including norms of the Jacobian at the origin). These conditions are chosen so that the classes properly contain the classical normalized holomorphic mappings while remaining strictly smaller than the full space of holomorphic mappings on the ball; non-emptiness is verified by explicit examples constructed in the same section. The lower bounds on the Bloch constants are obtained directly from these defining inequalities by applying a Bloch-radius argument adapted to several variables, without circularity. Comparisons with earlier results in the introduction show that the estimates are new and do not collapse to previously known cases. We agree that the abstract should state the defining conditions more explicitly and will revise it in the next version to include a concise description of the coefficient/growth restrictions. revision: yes

Circularity Check

0 steps flagged

No circularity: definitions of new classes yield non-trivial estimates via standard Bloch-radius techniques.

full rationale

The paper introduces two new classes of holomorphic mappings on B^n by explicit coefficient or growth conditions and applies known Landau-Bloch methods to obtain lower bounds on the Bloch constant. No step reduces the claimed estimates to the input definitions by construction, no fitted parameters are relabeled as predictions, and no load-bearing uniqueness result is imported solely via self-citation. The derivation chain remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated. The work presumably rests on standard axioms of several complex variables (Cauchy-Riemann equations, holomorphy on the ball) but none are enumerated.

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