A univalent function is a one-to-one function on its domain.

In modular function theory, a function is called univalent on a subgroup if it is automorphic under and has valence 1 (Apostol 1997, p. 84).

The univalent Bloch-Landau constant measures the largest disk guaranteed inside the image of a normalized univalent function, allowing the disk center to vary.

See also

Brennan's Conjecture, Clunie-Sheil-Small Conjecture, Universal Integral Means Spectrum, Valence

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References

Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 84, 1997.Das, S. "A Counterexample to the Clunie-Sheil-Small Coefficient-Difference Conjecture via Quasiconformal Maps." 6 Oct 2026. https://arxiv.org/abs/2610.07770.Wikström, F. "Bounds for the Univalent Bloch-Landau Constant." 25 Oct 2026. https://arxiv.org/abs/2610.34580.

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Univalent Function

Cite this as:

Weisstein, Eric W. "Univalent Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UnivalentFunction.html

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