Optimal Bloch constants for Bochner $(K,K')$-mappings and $(K,K')$-quasiregular mappings in the unit ball of $\mathbb{C}^n$

Allu and Kumar (Ann. Fenn. Math. 50 (2025), 215-229; arXiv:2308.15913) introduced two classes of holomorphic mappings of the unit ball $B^n\subset\mathbb{C}^n$ with an additive distortion parameter $K'$: Bochner $(K,K')$-mappings, satisfying $\|f'\|^2\le K^2|\det f'|^{2/n}+K'$ pointwise, and $(K,K')$-quasiregular mappings, satisfying $\Lambda_f\le K\lambda_f+K'$ for the extremal singular values of $f'$. They proved that normalized maps in these classes cover a schlicht ball of radius at least $R(n,K,K')=1/(4(K^2+K')^{n-1}(\sqrt{4K^2+K'}+\sqrt{K^2+K'}))$, respectively at least $\alpha^2/(4(2K\alpha+K'+\alpha))$ when $\lambda_f(0)\ge\alpha$, but established nothing about optimality. The problem is to determine the optimal quantitative Bloch theorem for these two classes: sharp lower bounds, the correct dependence of the constants on $n$, $K$, and $K'$ (including the decay as $K'\to\infty$), and extremal or asymptotically extremal mappings.

Background

Bloch's theorem in one complex variable states that a holomorphic function on the unit disc with $f'(0)=1$ maps some subdomain univalently onto a disc of radius at least an absolute constant $B>0$ (a schlicht ball is a ball covered one-to-one by the image of some subdomain, and the Bloch constant of a class is the largest radius guaranteed for all its normalized members). The exact value of the one-variable Bloch constant is a famous open problem: the Ahlfors-Grunsky conjecture predicts it, and even the best lower bound has only been inched upward (Bonk, 1990, as recalled in the source paper's introduction). In several complex variables the unrestricted problem degenerates: for holomorphic mappings $f:B^n\to\mathbb{C}^n$, $n\ge2$, there are normalized families whose images contain schlicht balls of arbitrarily small radius, so a positive Bloch constant exists only after restricting the class. Bochner (1946) introduced the class named after him: a holomorphic $f:B^n\to\mathbb{C}^n$ is a Bochner $K$-mapping if $\|f'\|\le K|\det f'|^{1/n}$ pointwise (here $\|f'\|=(\sum_{i,j}|a_{ij}|^2)^{1/2}$ is the Euclidean norm of the Jacobian matrix; following the source paper's convention, $\|\cdot\|$ always denotes this Euclidean norm, while $|A|=\sup_{z\ne0}|Az|/|z|$ is reserved for the operator norm), and proved that its normalized members have a positive Bloch constant without giving an explicit value. Takahashi (1951) treated the larger class defined by the ball-maximum inequality $\max_{|z|<r}\|f'(z)\|\le K\max_{|z|\le r}|\det f'(z)|^{1/n}$ and proved $\beta\ge(n-1)^{n-2}/(12K^{2n-1})$; Sakaguchi (1956) improved this to $\beta\ge(n-1)^{n-2}/(8K^{2n-1})$. Since a pointwise bound implies the ball-maximum bound, every Bochner $K$-mapping is a Takahashi $K$-mapping, so Sakaguchi's estimate also holds for Bochner's class. Another standard restriction is quasiregularity: writing $\Lambda_f(z)$ and $\lambda_f(z)$ for the largest and smallest singular values of $f'(z)$ (the square roots of the extreme eigenvalues of the Hermitian matrix $(f')^{*}f'$), a holomorphic $f$ is $K$-quasiregular if $\Lambda_f\le K\lambda_f$ pointwise; quasiregular holomorphic mappings are locally biholomorphic for $n>1$, and Chen and Gauthier (2000) proved explicit several-variable Bloch-type bounds for them. Graham and Varolin (1996) had earlier studied lower bounds for Bloch constants in one and several variables; in no several-variable class of this circle is a sharp constant known. Wang, Yang, and Liu (2022) proved a Landau-Bloch type theorem for $K$-quasiregular pluriharmonic mappings (mappings whose components are harmonic on every complex line), a model for the pluriharmonic variant. Allu and Kumar (2025) enlarged both classical classes by an additive parameter $K'\ge0$: a holomorphic $f:B^n\to\mathbb{C}^n$ is a Bochner $(K,K')$-mapping if $\|f'\|^2\le K^2|\det f'|^{2/n}+K'$ pointwise, and a $(K,K')$-quasiregular mapping if $\Lambda_f\le K\lambda_f+K'$ pointwise (for $K'=0$ these reduce to the classical classes; the definitions follow Nirenberg's planar idea of bounded additive perturbation). Their Theorem 1 shows that a Bochner $(K,K')$-mapping on $\overline{B^n}$ with $|\det f'(0)|=1$ maps some subdomain univalently onto a ball of radius at least $R(n,K,K')=\frac{1}{4(K^2+K')^{n-1}(\sqrt{4K^2+K'}+\sqrt{K^2+K'})}$; the case $K'=0$ gives $R(n,K,0)=\frac{1}{12K^{2n-1}}$ (Corollary 1 of the paper), which coincides with Takahashi's bound only up to the factor $(n-1)^{n-2}$ (exactly when $n=2$) and is provably non-sharp, since Sakaguchi's $\frac{(n-1)^{n-2}}{8K^{2n-1}}$ bound holds for the larger Takahashi class and exceeds $\frac{1}{12K^{2n-1}}$ for every $n\ge2$. Their Theorem 2 shows that a $(K,K')$-quasiregular mapping with $\lambda_f(0)\ge\alpha>0$ satisfies $\beta_f\ge\frac{\alpha^2}{4(2K\alpha+K'+\alpha)}$, where $\beta_f$ denotes the supremal radius of a schlicht ball in $f(B^n)$. The proofs combine a Takahashi-type rescaling argument, the Schwarz lemma for several complex variables, a singular-value estimate, and Hahn's univalence lemma, and at no point control the loss in these inequalities; no upper bound, no extremal example, and no optimality statement is given for either class.

Problem Statement

Determine the optimal quantitative Bloch theorem for the two mapping classes introduced by Allu and Kumar (2025) on the unit ball $B^n\subset\mathbb{C}^n$, $n\ge2$, $K\ge1$, $K'\ge0$. (a) For the class of Bochner $(K,K')$-mappings $f:\overline{B^n}\to\mathbb{C}^n$ (holomorphic with $\|f'(z)\|^2\le K^2|\det f'(z)|^{2/n}+K'$ pointwise) normalized by $|\det f'(0)|=1$: determine the sharp Bloch constant of the class, or at least the optimal dependence of any universal lower bound on $n$, $K$, and $K'$; in particular, decide whether the bound $R(n,K,K')=\frac{1}{4(K^2+K')^{n-1}(\sqrt{4K^2+K'}+\sqrt{K^2+K'})}$ is optimal (it is already known not to be sharp at $K'=0$), and construct mappings in the class that attain or asymptotically approach the true constant, thereby giving the first upper bounds. (b) For the class of $(K,K')$-quasiregular mappings $f:B^n\to\mathbb{C}^n$ (holomorphic with $\Lambda_f(z)\le K\lambda_f(z)+K'$ pointwise) with $\lambda_f(0)\ge\alpha>0$: decide whether $\beta_f\ge\frac{\alpha^2}{4(2K\alpha+K'+\alpha)}$ is optimal, and likewise determine the sharp constant or the optimal dependence on $(n,K,K',\alpha)$ with extremal or asymptotically extremal constructions. The common objective across both classes is the exact (or order-optimal) value of the guaranteed schlicht-ball radius, including how it decays as the additive distortion parameter $K'$ grows.

References

1. Vasudevarao Allu and Rohit Kumar, The Landau-Bloch type theorems for certain class of holomorphic and pluriharmonic mappings in $\mathbb{C}^n$, Annales Fennici Mathematici 50(1) (2025) 215-229, DOI 10.54330/afm.160790, https://doi.org/10.54330/afm.160790
2. Vasudeva Rao Allu and Rohit Kumar, The Landau-Bloch type theorems for certain class of holomorphic and pluriharmonic mappings in $\mathbb{C}^n$, arXiv:2308.15913 (v4, 2024), https://arxiv.org/abs/2308.15913
3. S. Bochner, Bloch's theorem for real variables, Bulletin of the American Mathematical Society 52 (1946) 715-719, DOI 10.1090/s0002-9904-1946-08640-1, https://doi.org/10.1090/s0002-9904-1946-08640-1
4. S. Takahashi, Univalent mappings in several complex variables, Annals of Mathematics 53 (1951) 464-471, DOI 10.2307/1969566, https://doi.org/10.2307/1969566
5. Koichi Sakaguchi, On Bloch's theorem for several complex variables, Science Reports of Tokyo Kyoiku Daigaku, Section A 5 (1956) 149-154, zbMATH Zbl 0073.30302, https://zbmath.org/?q=an%3A0073.30302
6. Huaihui Chen and P. Gauthier, Bloch constants in several variables, Transactions of the American Mathematical Society 353 (2000) 1371-1386, DOI 10.1090/s0002-9947-00-02734-3, https://doi.org/10.1090/s0002-9947-00-02734-3
7. Ian Graham and Dror Varolin, Bloch constants in one and several variables, Pacific Journal of Mathematics 174 (1996) 347-357, DOI 10.2140/pjm.1996.174.347, https://doi.org/10.2140/pjm.1996.174.347
8. Xin Wang, Yan Yang and Ming Sheng Liu, The Landau-Bloch type theorems for K-quasiregular pluriharmonic mappings, Monatshefte fur Mathematik 198(1) (2022) 189-209, DOI 10.1007/s00605-021-01657-y, https://doi.org/10.1007/s00605-021-01657-y
9. Shaolin Chen and David Kalaj, Conjugate type properties of harmonic $(K,K')$-quasiregular mappings, arXiv:2509.17578 (2025), https://arxiv.org/abs/2509.17578