Bloch constant and sharp one-variable-type constants for holomorphic mappings on unit balls of infinite-dimensional Banach spaces beyond the radial form $F(x)=f(x)x$

For holomorphic mappings on the unit ball of an infinite-dimensional complex Banach space, every currently known sharp analogue of classical one-variable geometric function theory — Bieberbach-type coefficient bounds, the Koebe one-quarter covering theorem, distortion theorems, the Fekete–Szegö inequality, Bohr–Rogosinski radii, and a lower bound for the Bloch constant — has been proved for the multiplicative radial class $F(x)=f(x)x$ (Hamada–Kohr–Kohr, 2024), with the Koebe one-quarter covering theorem since extended to normal Loewner chains on infinite-dimensional unit balls (Hamada–Kohr–Kohr, JFA 2026). This problem asks whether those sharp constants are universal for infinite-dimensional unit balls or artifacts of the radial structure: determine the Bloch constant (exact value or sharp bounds) for the radial class and for natural broader classes of Bloch mappings on infinite-dimensional balls, where the prenorm and determinant tools of finite-dimensional Bloch theory do not exist, and settle, by sharp extension or by explicit counterexample, the corresponding questions for the covering radius, coefficient bounds, and Bohr–Rogosinski radii, including their dependence on the geometry of the underlying Banach space.

Background

In one complex variable, geometric function theory revolves around a family of sharp constants for the class of normalized univalent (injective holomorphic) functions on the unit disc $U$: the Bieberbach conjecture, proved by de Branges (1985), states that the Taylor coefficients of $f(\zeta)=\zeta+\sum_{k\ge 2}a_k\zeta^k$ satisfy $|a_k|\le k$; the Koebe one-quarter theorem states that $f(U)$ contains the disc of radius $1/4$ around the origin; Bloch's theorem states that a holomorphic $f$ with $f'(0)=1$ maps some subdomain of $U$ biholomorphically onto a disc of radius bounded below by an absolute constant, and the best such constant $B$ (the Bloch constant) is famously unknown — Bonk (1990) proved $B\ge\sqrt3/4$, Chen–Gauthier (1996) improved this to $\sqrt3/4+2\cdot10^{-4}$, and Wikström (2026) further improved it to $\sqrt3/4+0.0153$, while the exact value remains open. Following the convention of the source paper, a schlicht ball of a mapping $F$ centered at $F(a)$ is a ball in the target with center $F(a)$ onto which $F$ maps some open subset of the domain containing $a$ biholomorphically; the Bloch constant of a class is the infimum over the class of the supremum of schlicht-ball radii. Related sharp constants include the Fekete–Szegö inequality (a sharp bound on a quadratic combination $|a_3-\lambda a_2^2|$ of Taylor coefficients), the Bohr radius (the largest $r$ such that $\sum|a_k|r^k\le\sup|f|$), and its refinement the Bohr–Rogosinski radius for functions subordinate to a univalent or convex function (computed in one variable by Kayumov–Khammatova–Ponnusamy, 2021); subordination $g\prec f$ means $g=f\circ\omega$ for some holomorphic $\omega:U\to U$ with $\omega(0)=0$. Alexander's theorem in one variable states that $f$ is convex if and only if $\zeta f'(\zeta)$ is starlike (starlike: $f(U)$ star-shaped about $0$; convex: $f(U)$ convex).

In several complex variables these statements fail for general mappings: already on the unit ball of $\mathbb C^2$ the shear map $(z_1,z_2)\mapsto(z_1+az_2^2,z_2)$ is normalized (vanishes at $0$ with identity derivative) and biholomorphic but violates the coefficient bound $|a_k|\le k$, and Alexander-type equivalences also fail (both facts are documented in the source paper, Hamada–Kohr–Kohr 2024). Positive multivariable theories therefore impose structure. For Bloch theory on the Euclidean ball $B^n$ of $\mathbb C^n$, Liu (1992) and FitzGerald–Gong (1994) established Bloch-type theorems, and Chen–Gauthier (2001) and Hamada (2019) obtained Bloch constant bounds, for the class of Bloch mappings normalized by $\|f\|_0=1$ and $\det Df(0)=1$, where $Df$ is the Fréchet derivative and $\|f\|_0=\sup_{z\in B^n}(1-\|z\|^2)^{c(B)/n}|\det Df(z)|^{1/n}$ is the prenorm ($c(B)$ a domain constant). These normalizations are essential to the finite-dimensional proofs, and both the prenorm and the determinant are meaningless in infinite-dimensional spaces: there is no determinant of a Fréchet derivative on an infinite-dimensional Banach space.

Hamada–Kohr–Kohr (arXiv:2409.04028, 2024) found a route around this obstruction for one structured class. On the unit ball $\mathbb B$ of an arbitrary complex Banach space $X$ (infinite-dimensional included), they study holomorphic mappings of the multiplicative radial form $F(x)=f(x)x$ with $f:\mathbb B\to\mathbb C$ holomorphic, $f(0)=1$. Their key univalence criterion reduces every question along complex lines $x=\zeta u$ ($\|u\|=1$) to the one-variable function $f_u(\zeta)=\zeta f(\zeta u)$, and they prove, all sharp within this class: the (generalized) Bieberbach bound $\|P_s\|\le s$ for the homogeneous expansion $F(x)=x+\sum_{s\ge2}P_s(x)$, answering positively their Question 1.1; the covering theorem $F(\mathbb B)\supseteq B_{1/4}$; sharp growth and (on Hilbert balls) sharp operator-norm distortion; a Fekete–Szegö inequality; a Bohr–Rogosinski radius for $G(x)=a+g(x)x$ subordinate to biholomorphic $F(x)=a+f(x)x$; an Alexander-type equivalence (quasi-convexity of type $B$ of $F$ $\iff$ starlikeness of $DF(x)x$); and, central to the present problem, a lower bound for the Bloch constant. Defining the Bloch semi-norm $\|F\|_B=\|F(0)\|+\sup_{x\ne0}(1-\|x\|^2)\|DF(x)x\|/\|x\|$ (which uses only the radial derivative and hence makes sense in infinite dimensions) and the class $\mathcal B_1(X)=\{F(x)=f(x)x: f(0)=1,\ \|F\|_B=1\}$, their Theorem 3.14 gives $B^*_X:=\inf\{r(F):F\in\mathcal B_1(X)\}\ge\sqrt3/4$, by lifting Bonk's one-variable distortion and covering argument through the univalence criterion.

Everything in this theory is thus conditioned on the factorization $F(x)=f(x)x$, which makes the line-reduction possible. The paper poses no formal open problems beyond this limitation, but its authors explicitly note that the prenorm and $\det Df(0)$ cannot be used in infinite dimensions — the reason only the radial class is treated — and the sharp constants within the radial class do not transfer automatically to any broader class. Even the one-variable Bloch constant is open, and every slice $f_u$ of an $F\in\mathcal B_1(X)$ belongs to the classical one-variable Bloch class, so the infinite-dimensional constants are tightly coupled to the classical problem. Whether a positive Bloch constant, a Koebe-type covering constant, Bieberbach-type coefficient bounds, or Bohr–Rogosinski radii exist at all for natural classes of mappings without the radial factorization — such as all Bloch mappings with $DF(0)=I$ and $\|F\|_B\le1$, or all normalized biholomorphic mappings — is unknown in general, the Bieberbach-type coefficient bounds being the case already decided — negatively, on the full normalized biholomorphic class, by the shear maps above: the only positive results beyond the radial form concern Loewner-chain subclasses (the Koebe one-quarter theorem for normal Loewner chains on unit balls of complex Banach spaces, Hamada–Kohr–Kohr, JFA 2026, and sharp Fekete–Szegö refinements for mappings embeddable as first elements of $g$-Loewner chains, Xu, 2026 — both proper subclasses of all normalized biholomorphic mappings), and no other counterexample families excluding further extensions are known either. For scalar-valued functions on Banach balls the Bohr phenomenon itself is understood (Hamada–Honda–Mizota, 2020, and later refinements), which highlights that the open frontier is precisely the vector-valued, univalence-based theory.

Problem Statement

Let $X$ be a complex Banach space of infinite dimension (in particular a complex Hilbert space $\mathcal H$), $\mathbb B\subset X$ its open unit ball, and $B_r\subset X$ the ball of radius $r$ centered at the origin. For $F\in H(\mathbb B,X)$ let $DF$ denote the Fréchet derivative, define the Bloch semi-norm $\|F\|_B=\|F(0)\|+\sup_{x\ne0}(1-\|x\|^2)\|DF(x)x\|/\|x\|$, and, following the source convention, let $r(F)$ be the supremum over $a\in\mathbb B$ of the radii of schlicht balls of $F$ centered at $F(a)$ (balls in $X$ with center $F(a)$ onto which $F$ maps some open subset of $\mathbb B$ containing $a$ biholomorphically). The source paper proves, for the multiplicative radial class $\mathcal B_1(X)=\{F(x)=f(x)x:\ f\in H(\mathbb B,\mathbb C),\ f(0)=1,\ \|F\|_B=1\}$, that $B^*_X:=\inf\{r(F):F\in\mathcal B_1(X)\}\ge\sqrt3/4$, along with sharp covering ($1/4$), Bieberbach ($\|P_s\|\le s$ for the homogeneous expansion), distortion, Fekete–Szegö, and Bohr–Rogosinski constants, all via a line-reduction available only because of the factorization $F(x)=f(x)x$. Determine, with the Bloch constant as the central case, how far these sharp one-variable-type constants extend beyond that factorization on infinite-dimensional unit balls:

1. For the radial class $\mathcal B_1(X)$ itself: improve or determine exactly the Bloch constant $B^*_X$ (currently $B^*_X\ge\sqrt3/4$, exact value open and coupled to the open one-variable Bloch constant), including sharp upper bounds via explicit extremal families.
2. For classes without the radial factorization — at minimum the class of Bloch mappings $F\in H(\mathbb B,X)$ with $F(0)=0$, $DF(0)=I$, $\|F\|_B\le1$ on an infinite-dimensional ball, and the class of normalized biholomorphic mappings of $\mathbb B$: does a positive Bloch constant exist? If yes, give sharp lower and upper bounds (or the exact value); if no, construct an explicit family in the class with $r(F)\to0$.
3. For the same two broader classes (each constant below is asked on the class named with it): determine the covering constant on the class of normalized biholomorphic mappings of $\mathbb B$ (is there a Koebe-type radius $\rho>0$ with $F(\mathbb B)\supseteq B_\rho$ for all such $F$?), the Bieberbach-type coefficient bounds $\|P_s\|\le c_s$ on the Bloch class $\{F(0)=0,\ DF(0)=I,\ \|F\|_B\le1\}$ (to which the shear counterexample recorded in the Background does not apply), the Fekete–Szegö-type inequalities on both classes, and the Bohr–Rogosinski radii for mappings subordinate to either class — each either with its sharp value or with an explicit counterexample family showing no such constant exists. Partial extensions beyond the radial class are known for Loewner-chain subclasses: the Koebe one-quarter theorem holds for normal Loewner chains on the unit ball of a complex Banach space (Hamada–Kohr–Kohr, JFA 2026), and sharp Fekete–Szegö refinements hold for normalized biholomorphic mappings embeddable as first elements of $g$-Loewner chains (Xu, 2026); the open question is the full class of normalized biholomorphic or Bloch mappings beyond these chain subclasses.
4. Clarify how the answers to 1–3 depend on the geometry of $X$, in particular Hilbert balls versus balls of non-Hilbert Banach spaces (the sharp operator-norm distortion theorem of the source paper is already available only on Hilbert balls).

The four items together constitute the problem.

References

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