Sharpness of the Bloch-type ratio bound for quasiregular pluriharmonic mappings on homogeneous unit balls

For a pluriharmonic mapping $f=h+\bar g$ with dilatation bound $\|\omega_f\|\le k$ on a homogeneous unit ball of $\mathbb{C}^n$, Theorem 2.4 of Chen–Hamada–Ponnusamy–Vijayakumar (J. d'Analyse Math. 152 (2024) 181–216) bounds the ratio of univalent-ball radii $B_f(z)/B_h(z)$ between $1-k$ and $\mu_k(\|\omega_f(z)\|/k)\le 1+k$, where $\mu_k(x)=1+k(1/x+(1-1/x^2)\log(1+x))$ and $\mu_k(0)=1+k/2$. The lower bound is sharp, but the upper bound is proved sharp only asymptotically as $k\to 0^+$. The open problem: for each fixed $k\in(0,1)$, determine whether $\mu_k$ is the optimal upper function and identify the extremal mappings, and, more broadly, determine the exact Bloch-type constants of constrained pluriharmonic and holomorphic classes on homogeneous balls and their Banach/JB*-triple extensions.

Background

Classically, Bloch's theorem states that a holomorphic function $f$ on the unit disk with $|f'(0)|=1$ maps some subdomain univalently (that is, one-to-one) onto a Euclidean disk of uniformly bounded radius. The largest such uniform radius is the Bloch constant $\mathcal B$, and its exact value has been open since the 1920s: the Ahlfors–Grunsky upper bound $\mathcal B\le 0.471861\ldots$ is conjectured to be exact, while the best lower bounds (Ahlfors 1938, with numerical improvements by Bonk 1990 and Chen–Gauthier 1996) remain near $\sqrt 3/4\approx 0.433$. In several complex variables no absolute Bloch constant exists for unconstrained holomorphic maps — for example $f_m(z)=(mz_1,z_2/m,z_3,\ldots,z_n)$ maps the unit ball univalently while its image contains no ball of radius larger than $1/m$ — so one must constrain the class of mappings.

The source paper studies a natural constrained class. A mapping $f$ from a domain in $\mathbb{C}^n$ into $\mathbb{C}^n$ is pluriharmonic if $f=h+\bar g$ with $h$ and $g$ holomorphic (the bar denotes componentwise conjugation); writing $D$ for the Fréchet derivative, the second complex dilatation is the matrix-valued function $\omega_f=Dg[Dh]^{-1}$. When $\|\omega_f\|\le k<1$ (operator norm), $f$ is quasiregular; in the planar case this is exactly a $K$-quasiregular harmonic mapping with $K=(1+k)/(1-k)$. Let $B$ be the homogeneous unit ball of $\mathbb{C}^n$ with respect to some norm — equivalently, the unit ball of a finite-dimensional JB*-triple, i.e. a complex Banach space with a Jordan triple product whose open unit ball is homogeneous under biholomorphic self-maps; this class contains the Euclidean ball, the polydisc, and balls of rectangular matrix spaces. Denote by $\mathcal{PH}(k)$ the pluriharmonic mappings $f=h+\bar g:B\to\mathbb{C}^n$ with $h$ locally biholomorphic, $h(0)=g(0)=0$, and $\|\omega_f\|\le k$. For $f\in\mathcal{PH}(k)$ and $a\in B$, $B_f(a)$ is the radius of the largest Euclidean ball centered at $f(a)$ that is the univalent image of some open subset of $B$ containing $a$ (a schlicht ball), and $B_h(a)$ is the same quantity for the holomorphic part $h$.

Because an absolute Bloch-type constant depends on the choice of normalization and vanishes for unconstrained classes in several variables, the source paper measures $f$ against its holomorphic part, which needs no normalization: Theorem 2.4 of Chen–Hamada–Ponnusamy–Vijayakumar proves that for $f=h+\bar g\in\mathcal{PH}(k)$ and $z\in B$,

$$1-k\;\le\;\frac{B_f(z)}{B_h(z)}\;\le\;\mu_k\!\left(\frac{\|\omega_f(z)\|}{k}\right)\;\le\;\mu_k(1)=1+k,$$

where $\mu_k(x)=1+k\bigl(1/x+(1-1/x^2)\log(1+x)\bigr)$ for $x\in(0,1]$ and $\mu_k(0)=1+k/2$. The left inequality is sharp for every $z$, attained by the affine maps $f(z)=Uz+k\,\overline{Uz}$ with $U$ unitary, but the right inequality is established only asymptotically as $k\to 0^+$ (where $\mu_k\to 1$): for each fixed nonzero $k$, neither the optimality of $\mu_k$ nor the extremal mappings are known. The same paper also proves Schwarz–Pick type estimates for pluriharmonic mappings on JB*-triples, but those are pointwise sharp; the unresolved questions are the fixed-$k$ sharpness of the ratio bound above and the exact values of Bloch-type constants, which the paper's introduction lists among the long-standing open problems of the field.

Problem Statement

Determine the sharp upper bound in the Bloch-type ratio inequality of Theorem 2.4 of Chen–Hamada–Ponnusamy–Vijayakumar, and the exact Bloch-type constants, for quasiregular pluriharmonic mappings on homogeneous unit balls. Concretely:

(i) For each fixed $k\in(0,1)$ and each homogeneous unit ball $B\subset\mathbb{C}^n$ (in particular the Euclidean ball), determine the sharp constant

$$\Lambda_k(x)=\sup\left\{\frac{B_f(z)}{B_h(z)}:\; f=h+\bar g\in\mathcal{PH}(k),\; z\in B,\; \frac{\|\omega_f(z)\|}{k}=x\right\},\qquad x\in[0,1],$$

where $\mathcal{PH}(k)$ is the class of pluriharmonic mappings $f=h+\bar g:B\to\mathbb{C}^n$ with $h$ locally biholomorphic, $h(0)=g(0)=0$, and $\|\omega_f\|\le k$ for $\omega_f=Dg[Dh]^{-1}$, and $B_f(a)$ denotes the radius of the largest Euclidean ball centered at $f(a)$ that is the univalent image of an open subset of $B$ containing $a$. The target is the exact function $\Lambda_k(x)$ on $[0,1]$, and in particular whether $\Lambda_k(x)=\mu_k(x)$ with $\mu_k(x)=1+k(1/x+(1-1/x^2)\log(1+x))$ for $x\in(0,1]$ and $\mu_k(0)=1+k/2$ — i.e. whether the upper bound of Theorem 2.4 is optimal for fixed $k$. Identify the extremal mappings. (The lower bound $1-k$ is sharp and attained by affine maps, so the question concerns only the upper side.)

(ii) More generally, determine the exact Bloch-type constant $\mathcal B(\mathcal P)=\inf_{f\in\mathcal P}\sup_{a\in B}B_f(a)$ for normalized subclasses $\mathcal P$ — subclasses with $\|Dh(0)\|=1$ — of the class $\mathcal{PH}(k)$ on homogeneous unit balls of $\mathbb{C}^n$ and of the constrained holomorphic classes on homogeneous unit balls. In particular: (a) determine the exact value of $\mathcal B(\mathcal P)$ for the Euclidean-ball instance $\mathcal P=\{f=h+\bar g\in\mathcal{PH}(k):\|Dh(0)\|=1\}$. (b) For every other normalized subclass of $\mathcal{PH}(k)$, for the constrained holomorphic classes on homogeneous unit balls, and for the analogues on unit balls of infinite-dimensional Banach or JB*-triple spaces, determine the exact value of $\mathcal B(\cdot)$, and in particular whether $\mathcal B(\cdot)>0$.

The two parts (i) and (ii) share one scientific objective: replace the known two-sided bounds by exact constants with identified extremals in the Bloch-type theory of constrained (quasiregular) pluriharmonic mappings. Part (i) is the normalization-free core left open by the source theorem; part (ii) is the absolute-constant formulation the source situates it in, whose special cases include the classical one-variable Bloch constant.

References

1. Shaolin Chen, Hidetaka Hamada, Saminathan Ponnusamy, Ramakrishnan Vijayakumar, Schwarz type lemmas and their applications in Banach spaces, Journal d'Analyse Mathématique 152 (2024) 181–216, DOI 10.1007/s11854-023-0293-0 (preprint arXiv:2110.02767), https://doi.org/10.1007/s11854-023-0293-0
2. Shaolin Chen, Hidetaka Hamada, Megha Kundathil, Ramakrishnan Vijayakumar, Some sharp Schwarz type estimates and their applications in Banach spaces, arXiv:2605.17237 (2026), https://arxiv.org/abs/2605.17237
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