The Hayman--Wu constant is $π^2$

Abstract: We show that for every conformal map $φ:\mathbb{D}\toΩ\subsetneq\mathbb{C}$ from the unit disk onto a simply connected proper domain $Ω$, the length of $φ^{-1}(Ω\cap L)$ is at most $π<sup>2$ for every line $L$.

• The paper proves that the sharp Hayman–Wu constant is exactly π², establishing the upper bound and matching Öyma’s constructions approaching this value.
• A contact-selector lemma for univalent maps from the upper half-plane to the disk provides the optimal π/2 factor by controlling vertical image length through selected boundary length.
• Reflection across the line, followed by analytic Jordan-domain approximation and radial exhaustion, extends the result to all simply connected proper domains without boundary regularity assumptions.

The paper proves the sharp form of the Hayman–Wu theorem: if $\phi:\mathbb D\to\Omega\subsetneq\mathbb C$ is conformal and $L$ is any Euclidean line, then the conformal length of the preimage of the line segment inside $\Omega$ satisfies

$$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$

Together with the lower-bound constructions of Öyma, this establishes that the Hayman–Wu constant is exactly $\pi^2$ (2608.12844). The central contribution is a contact-selector inequality for univalent functions on the upper half-plane. This estimate supplies the missing sharp factor in the reflected-domain proof of the theorem.

The sharp Hayman–Wu problem

For a conformal map $\phi:\mathbb D\to\Omega$ and a line $L$, the quantity of interest is the Euclidean length, measured in the disk, of

$\phi^{-1}(\Omega\cap L).$

The geometry in the image domain can be arbitrarily complicated: a line may intersect a simply connected domain in many components, while the conformal preimage need not have a simple Euclidean description. The Hayman–Wu theorem asserts that this preimage length is uniformly bounded over all simply connected proper domains, conformal maps, and lines.

The optimal constant was previously bracketed by successive estimates. Hayman and Wu established finiteness; Öyma obtained the upper bound $4\pi$; Rohde improved this to a strict inequality below $4\pi$; and Öyma constructed examples whose conformal lengths tend to $L$0. The paper closes the gap by proving the upper bound $L$1, thereby matching the lower asymptotic construction. The result is consequently sharp but, as the construction indicates, sharpness need not be attained by a regular extremal configuration.

The proof is based on reflection across the line. After a Euclidean motion, the line is taken to be the real axis. The domain is intersected with its reflection across that axis, and each component intersecting the real line is controlled separately. The conformal length of the real-axis portion is then related to the boundary length of a selected portion of the reflected component. The global estimate follows because the selected boundary pieces are disjoint and collectively have length at most the circumference of the unit circle.

The contact-selector lemma

The main new result is a sharp estimate for a univalent map $L$2, where $L$3 denotes the upper half-plane. Let $L$4 be a finite union of disjoint open intervals satisfying the selector condition that, for almost every $L$5, exactly one of $L$6 and $L$7 belongs to $L$8. Suppose further that $L$9 extends holomorphically across $\Omega$0 and maps $\Omega$1 to the unit circle. The lemma states that

$\Omega$2

The factor $\Omega$3 is optimal (2608.12844).

This inequality compares the length of the image of the positive imaginary axis with the image length of a boundary selector that chooses one member from almost every reflected pair $\Omega$4. Its significance is structural: the vertical image corresponds to the line intersection in the reflected-domain reduction, while the selected real intervals correspond to portions of the original domain boundary. The $\Omega$5 factor is therefore precisely the local constant required to combine the componentwise estimate with the global boundary-length bound $\Omega$6.

Factorization and positivity

The proof factors $\Omega$7 into a possible Blaschke factor and a zero-free factor: $\Omega$8 Since $\Omega$9 is univalent, it has at most one zero in $$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$0. If $$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$1 is that zero, then

$$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$2

otherwise $$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$3. The remaining factor $$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$4 is zero-free and bounded by one in modulus. It is written as

$$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$5

where $$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$6 is either constant real-valued or a Pick function with nonnegative imaginary part.

Because $$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$7 extends across every interval of $$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$8 and has unimodular boundary values there, $$\operatorname{length}\bigl(\phi^{-1}(\Omega\cap L)\bigr)\leq \pi^2.$$9 extends holomorphically across those intervals and is real-valued on them. Its Herglotz representation is

$\pi^2$0

with $\pi^2$1 and $\pi^2$2 a positive measure. The absence of singular measure on intervals through which $\pi^2$3 extends yields

$\pi^2$4

The selector set has infinite Lebesgue measure. Consequently, finiteness of $\pi^2$5 forces $\pi^2$6. This observation is important: the affine term in the Herglotz representation would contribute a positive constant to the boundary derivative and would make the selected boundary integral divergent. Thus all relevant variation is carried by the representing measure $\pi^2$7 and, when present, by the single Blaschke zero.

On the selected boundary intervals, the logarithmic derivative has no cancellation: $\pi^2$8 where

$\pi^2$9

if $\phi:\mathbb D\to\Omega$0 has the zero $\phi:\mathbb D\to\Omega$1, and $\phi:\mathbb D\to\Omega$2 otherwise. This exact additivity is a decisive feature of the argument. It reduces the estimate to separate bounds for the zero-free factor and the possible zero.

The zero-free factor

Schwarz–Pick applied to the Pick function $\phi:\mathbb D\to\Omega$3 gives

$\phi:\mathbb D\to\Omega$4

Integrating along the imaginary axis and applying Tonelli's theorem yields

$\phi:\mathbb D\to\Omega$5

The selector property supplies the complementary lower bound. For every $\phi:\mathbb D\to\Omega$6,

$\phi:\mathbb D\to\Omega$7

Indeed, for each positive pair $\phi:\mathbb D\to\Omega$8, the selector chooses one of the two kernels $\phi:\mathbb D\to\Omega$9 and $L$0, and the smaller of these kernels already has integral $L$1 over the positive half-line when $L$2. Integrating this estimate against $L$3 gives

$L$4

Therefore,

$L$5

The implication is that the entire vertical variation of the zero-free factor is controlled by the selected boundary variation with the sharp factor $L$6. No geometric regularity beyond the hypotheses of the lemma is used at this stage.

The possible Blaschke zero

The Blaschke factor is treated explicitly. Along the imaginary axis,

$L$7

when $L$8. The selector condition gives

$L$9

The elementary inequality

$\phi^{-1}(\Omega\cap L).$0

then implies

$\phi^{-1}(\Omega\cap L).$1

The case $\phi^{-1}(\Omega\cap L).$2 is checked separately and satisfies the same inequality.

Since $\phi^{-1}(\Omega\cap L).$3, one has

$\phi^{-1}(\Omega\cap L).$4

Combining the estimates for $\phi^{-1}(\Omega\cap L).$5 and $\phi^{-1}(\Omega\cap L).$6 with the exact boundary identity $\phi^{-1}(\Omega\cap L).$7 proves the contact-selector lemma.

The lemma's constant is not merely inherited from a coarse inequality. Its sharpness is demonstrated by taking $\phi^{-1}(\Omega\cap L).$8 to be a single Blaschke factor with zero $\phi^{-1}(\Omega\cap L).$9 and selecting $4\pi$0. As $4\pi$1,

$4\pi$2

so the quotient tends to $4\pi$3. This identifies the local extremal mechanism: a zero approaching a boundary direction far from the selected half-line asymptotically saturates the selector inequality.

Reflection and the analytic Jordan case

The geometric argument is first established for bounded analytic Jordan domains. Let $4\pi$4 be conformal, and define the reflected domain

$4\pi$5

Let $4\pi$6 denote the components of $4\pi$7, and let $4\pi$8 be the component of $4\pi$9 containing $4\pi$0. Standard planar separation properties imply that each $4\pi$1 is a Jordan domain symmetric about the real axis, that

$4\pi$2

and that distinct components have boundary intersections contained in the real axis, with at most one common point.

For each $4\pi$3, symmetry and Carathéodory extension produce a conformal map

$4\pi$4

satisfying

$4\pi$5

The boundary of $4\pi$6 consists of arcs lying on either $4\pi$7 or $4\pi$8. The real axis is partitioned by the preimages of the intersection points $4\pi$9 and by the point at infinity. On each resulting interval, Schwarz reflection gives holomorphic continuation of $L$00, and the interval maps entirely into one of the two reflected boundaries.

For each reflected pair of intervals, the construction selects the member mapped into $L$01. If both members map into $L$02, the positive interval is selected. Denote the resulting selector by $L$03, its image by

$L$04

and set

$L$05

The function $L$06 satisfies the hypotheses of the contact-selector lemma: it is univalent into $L$07, extends across $L$08, and has unimodular boundary values there. Hence

$L$09

The sets $L$10 are pairwise disjoint except possibly at endpoints. Since $L$11 extends homeomorphically from $L$12 onto $L$13,

$L$14

The components $L$15 are disjoint and exhaust $L$16. Summing the component estimates therefore gives

$L$17

The numerical constant has a transparent decomposition: $L$18 is the sharp analytic selector constant, while $L$19 is the total boundary length of the unit circle. The result is thus obtained without an additional loss in either the local analytic estimate or the global geometric summation.

Passage to arbitrary simply connected domains

The general case is obtained by radial exhaustion. After reducing the line to the real axis, let

$L$20

where $L$21. For $L$22, the domain $L$23 is a bounded analytic Jordan domain, so the preceding argument gives

$L$24

The relevant preimage sets satisfy

$L$25

and increase as $L$26. Continuity from below of Lebesgue measure therefore yields

$L$27

This step removes boundedness and boundary regularity assumptions without requiring direct boundary extension for the original conformal map. The argument uses only the regularity of the inner approximants and monotone convergence of the resulting Borel subsets of the disk.

Öyma's examples with conformal lengths tending to $L$28 show that this upper bound cannot be reduced. The supremum defining the Hayman–Wu constant is consequently

$L$29

Limitations and open questions

The proof establishes sharpness at the level of the supremum, but it does not identify an extremal conformal map and line for which equality is attained. The lower-bound examples only produce sequences approaching $L$30. Whether equality can occur under additional geometric hypotheses, or whether every exact extremizer must degenerate in a specific way, remains outside the paper's scope.

The analytic Jordan-domain argument also relies on the finite intersection structure of $L$31 and its reflection. This is appropriate for the approximation step, but the proof does not provide a direct geometric description of selector sets for arbitrary rough boundaries. The radial exhaustion bypasses that issue rather than resolving it intrinsically.

Finally, the contact-selector lemma is formulated for finite unions of intervals. Although this is sufficient for the analytic approximation and the limiting theorem, an extension to more general measurable selectors could clarify which features of the interval decomposition are essential and whether analogous sharp inequalities hold for broader boundary-contact configurations.

Conclusion

The paper proves the sharp Hayman–Wu inequality

$L$32

for every conformal map from the disk onto a simply connected proper domain and every line $L$33 (2608.12844). Its principal technical result is the sharp $L$34 contact-selector estimate for univalent maps of the upper half-plane into the disk. Reflection across the line converts each interior line component into a comparison with a selected boundary set, and disjointness of those sets supplies the global factor $L$35. The product of the two factors gives $L$36, matching Öyma's lower constructions and determining the Hayman–Wu constant exactly.