A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius

Mikhail Borovikov

Home/Archive/arXiv

arXiv · 2609.39676

A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius

Mikhail Borovikov

View original source ↗Read source document ↗Export metadata ↓

Abstract

We disprove a conjecture of Liu, Luo, and Luo (2020) concerning the Landau radius for bounded planar harmonic mappings. We construct an explicit harmonic mapping $F_M$ satisfying the standard normalization conditions, which fails to be locally univalent strictly within the classical holomorphic Landau disk for all $M > M^* \approx 4.451$. By combining this counterexample with the lower bound established by Ponnusamy, Kalaj, and Vuorinen (2014), we demonstrate that the exact asymptotics of the harmonic Landau radius is $\fracπ{8M}$.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Mikhail Borovikov. 2026-09-30. A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius. https://arxiv.org/abs/2609.39676

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Second Main Theorem for Entire Curves Intersecting Three Conics

We establish a Second Main Theorem for algebraically nondegenerate entire holomorphic curves $ f: \mathbb{C} \to \mathbb{P}^2 $ intersecting a generic configuration of three conics $\mathcal{C}= \mathcal{C}_1+ \mathcal{C}_2+ \mathcal{C}_3 $ in the complex projective plane $\mathbb{P}^2$. Using invariant logarithmic $2$-jet differentials with negative twists, we prove the estimate \[ T_f(r) \leqslant 5 \sum_{i=1}^3 N_f^{[1]}(r, \mathcal{C}_i) + o\big(T_f(r)\big)\quad\parallel, \] where $ T_f(r) $ is the Nevanlinna characteristic function, and $ N_f^{[1]}(r, \mathcal{C}_i) $ is the $1$-truncated counting function. The key innovation of our approach is establishing new vanishing lemmas of the form \[ H^0\bigl(\mathbb{P}^2,\, E_{2,m}T_{\mathbb{P}^2}^*(\log \mathcal{C}) \otimes \mathcal{O}_{\mathbb{P}^2}(-t)\bigr) = 0 \] for specific pairs $(m, t)$, achieved by combining algebro-geometric arguments with computer-assisted computations through a mod-$p$ reduction technique. This provides a systematic method for proving vanishing results for negatively twisted jet differentials--a key component in complex hyperbolic geometry--thereby filling a missing piece in two-dimensional complex hyperbolicity theory.

math.CV↗

arXiv2026

Vanishing of Invariant 2-Jet Differentials and Improved Hyperbolicity Degree Bounds in Dimension Two

This paper establishes new degree bounds for Kobayashi hyperbolicity in dimension two. Our main results are: -- A very generic surface in $\mathbb{P}^3$ of degree at least $17$ is Kobayashi hyperbolic. -- The complement of a very generic curve in $\mathbb{P}^2$ of degree at least $12$ is Kobayashi hyperbolic. These bounds improve the long-standing records in the field, lowering the threshold from $18$ to $17$ for surfaces (Păun) and from $14$ to $12$ for complements (Rousseau). Central to the proofs are new vanishing results for certain negatively twisted invariant $2$-jet differentials, obtained through a novel combination of algebraic reduction and computer algebra. Since Demailly's Santa Cruz lectures in 1995, the thresholds for the existence of such differentials---and consequently the limits of what $2$-jet techniques can accomplish toward the Kobayashi conjecture in dimension two---have been recognized as $d = 15$ in the compact case and $d = 11$ in the logarithmic case. While previous approaches were unable to reach these targets, the present work provides both the theoretical foundations and the algorithmic framework required to access them, and has already improved the known bounds to $d = 17$ and $d = 12, 13$, respectively. As an unexpected byproduct, our computational method reveals the existence of nonzero negatively twisted invariant $2$-jet differentials with weighted degree $3$ for hyperelliptic-type equations of degree at least $11$ in the logarithmic case and degree at least $15$ in the compact case. Moreover, in the logarithmic setting, we establish an effective quantitative refinement via a Second Main Theorem in Nevanlinna theory.

math.CV↗

arXiv2026

Logarithmic coefficients for exponential classes of starlike and convex functions

In this paper, we investigate two subclasses of analytic and univalent functions associated with the exponential mapping $φ(z)=e^{αz},\qquad 0<α\le1,$ defined via the subordination conditions $\frac{zf'(z)}{f(z)}\prec e^{αz} \quad \text{and} \quad 1+\frac{zf''(z)}{f'(z)}\prec e^{αz}$. These classes provide a natural exponential analogue of several classical subclasses arising in geometric function theory. We obtain sharp coefficient estimates, logarithmic coefficient inequalities and sharp bounds for the associated Hankel and upper bounds for Toeplitz determinants. In particular, explicit estimates are derived for $$ |H_{2,1}(F_f/2)|, \quad |T_{2,1}(F_f/2)|, $$ for functions belonging to the introduced exponential subclasses of starlike and convex functions. Our results extend and unify several earlier works on exponential subclasses and highlight connections with logarithmic coefficients and determinant functionals.

math.CV↗