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Harmonic Landau Radius

The harmonic Landau radius r_(harm)(M), for M>1, is the supremum of the numbers r>0 such that every complex-valued harmonic function F=h+g^_ on the unit disk satisfying F(0)=0, lambda_F(0)=1, and |F(z)|<M is univalent on |z|<r, where h and g are analytic functions and

lambda_F(z)=|F_z(z)|-|F_(z^_)(z)|=|h^'(z)|-|g^'(z)|.

Liu et al. (2020) conjectured that r_(harm)(M)=1/(M+sqrt(M^2-1)). Their theorem under the stronger hypothesis |h(z)|+|g(z)|<M does not imply this assertion under the bound |F(z)|<M.

Kalaj et al. (2014) gave a lower bound for r_(harm)(M). Borovikov (2026) constructed explicit bounded complex-valued harmonic functions whose jacobian determinants vanish before the classical holomorphic Landau radius whenever M>M^*=(16-2pi+pi^(3/2))/[4(4-pi)]=4.451.... The upper bound from these counterexamples and the earlier lower bound have the same leading term, proving

r_(harm)(M)∼pi/(8M)

as M->infty (Borovikov 2026).

The harmonic Landau radius concerns a guaranteed disk of univalence for normalized bounded complex-valued harmonic functions and is distinct from the Landau constant, which concerns disks contained in images of normalized analytic functions.

See also

Harmonic Function, Landau Constant, Univalent Function

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References

Borovikov, M. "A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius." 30 Sep 2026. https://arxiv.org/abs/2609.39676.Kalaj, D.; Ponnusamy, S.; and Vuorinen, M. "Radius of Close-to-Convexity and Fully Starlikeness of Harmonic Mappings." Complex Var. Elliptic Equ. 59, 551-564, 2014.Liu, M.-S.; Luo, L.-F.; and Luo, X. "Landau-Bloch Type Theorems for Strongly Bounded Harmonic Mappings." Monatsh. Math. 191, 175-185, 2020.

Cite this as:

Weisstein, Eric W. "Harmonic Landau Radius." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarmonicLandauRadius.html

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