Alec Kriebel

Nonmonotone Argument and Radial Angle of a Support Slit

By Alec Kriebel, independent researcher.
Manuscript and declared publication date · Zenodo version 1.0 · Unrefereed preprint

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Original abstract

We give negative answers to the two slit-monotonicity questions of P. L. Duren recorded as Problem 6.72 in Hayman–Lingham. Every global maximizer of the real part of a₂+i10⁻²⁴(a₄−3a₃) in the normalized schlicht class omits an arc with nonmonotone continuous argument and nonmonotone radial angle, under both signed and ordinary unsigned conventions. The proof uses classical support-point geometry, a quantitative area-theorem estimate for the actual slit tip, and controlled inversion of the Schiffer differential. It establishes actual extremizers by compactness and does not assume that an arbitrary trajectory is an extremal slit. No formula for the maximizing function or uniqueness is required. This is an unrefereed AI-assisted research note.

The idea in plain English

A univalent function maps the unit disk into the complex plane without folding it over itself. A support point is such a map that maximizes a linear measurement of the function. Classical results show that its image leaves out a single analytic arc, running from a finite tip to infinity.

Two questions attributed to P. L. Duren ask whether that omitted arc must turn monotonically in its polar direction, and whether the angle between its radius and its tangent must change monotonically. Here monotone allows either increasing or decreasing behavior.

This note chooses one particular measurement of the second, third and fourth Taylor coefficients. Global maximizers exist, and every one of them has an actual omitted slit for which both proposed monotonicity properties fail. The radial-angle failure holds for both the signed outward angle and the ordinary nonnegative angle.

The proof controls all maximizers at once. It does not need to write down a maximizing function or show that it is unique. The polar direction, the tangent direction and the angle between them are different quantities; the theorem addresses the first and the third.

The precise result

Let S be the analytic injective maps on the unit disk normalized by f(0) = 0 and f′(0) = 1, with f(z) = z + Σₙ≥₂ aₙ(f)zⁿ. Set ε = 10⁻²⁴ and L(f) = a₂(f) + iε(a₄(f) − 3a₃(f)). Global maximizers of Re L on S exist. Every such maximizer omits an actual analytic slit Γ whose continuous polar argument, signed outward radial angle and ordinary unsigned outward radial angle are all nonmonotone.

The polar argument is a continuous real-valued lift along Γ. The tangent used for the signed radial angle points from the finite tip toward infinity. The ordinary radial angle is the nonnegative angle between that outward tangent and the radius vector. Both nondecreasing and nonincreasing behavior are excluded for each of the three quantities.

This supplies counterexamples to the two support-slit monotonicity properties recorded as Hayman–Lingham Problem 6.72. No explicit maximizing map or uniqueness theorem is required.

Why the result holds

Compactness of S gives genuine global maximizers. Comparing any maximizer with the Koebe function and using the classical coefficient inequalities gives uniform control of its first coefficients. An odd-root version of the area theorem then controls an analytic tail and places the actual omitted slit tip inside radius 3/10.

The Schiffer differential is expanded exactly for the chosen functional. A controlled inverse natural coordinate constructs a regular arc with modulus greater than 25/81. It agrees with the actual omitted slit near infinity. The separation from the actual tip and local trajectory uniqueness extend that agreement across the tested subarc. This identifies a subarc of each genuine support slit; it does not infer extremality from an arbitrary differential trajectory.

Writing this subarc as W(t) = −V(t)⁻², with 0 < t ≤ 7/4, and Z = tV′/V, the proof bounds Z − 1 − iεq(t) by 2000εη, where η = 10⁻¹² and q(t) = t² − 2t⁴/5. The values q(1) = 3/5 and q(7/4) = −441/640 force opposite strict signs for Im Z and hence opposite signs of the polar-argument derivative.

Increasing t points inward, so the outward tangent is −W′ and its signed radial angle is β = arg Z. This angle tends to zero at infinity and takes both positive and negative values; those facts exclude both monotone directions. Its continuity gives an interior zero between nonzero endpoint values, so the ordinary angle |β| is separately nonmonotone.

Scope and limitations

The theorem concerns the stated normalized schlicht class, the particular degree-four coefficient functional and the fixed value ε = 10⁻²⁴. It is not a conclusion about every coefficient functional or every support slit. At ε = 0 the strict reversal disappears; no claim for arbitrary larger perturbations is made.

The existence and every-maximizer conclusions do not provide an explicit maximizing function, uniqueness, a numerical extremal solver or a computed extremizer.

The proof uses established compactness, Bieberbach coefficient bounds, the area theorem and classical support-slit and Schiffer geometry. Finite rational checks and numerical diagnostics supplement the analytic estimates; they do not formally certify the universal theorem or the classical foundations.

The historical comparisons are bounded. The manuscript identifies later Duren commentary and Hibschweiler’s 1995 paper as uninspected. No historical firstness, exhaustive priority clearance or claim that the questions remained open worldwide in 2026 is made.

This is an unrefereed preprint developed with extensive AI assistance. Internal AI reviews are not independent human or journal peer review, and no formal proof-assistant certification is claimed.

Status and AI use

AI tools were used extensively in mathematical development, derivation, writing, reproduction, adversarial verification and subsequent review. Internal AI review is not independent human or journal peer review. This is an unrefereed preprint without formal proof-assistant certification. The named author is responsible for its claims, materials and citations.

The manuscript and Zenodo’s declared publication date are 14 October 2026. Zenodo created the record on 15 October 2026. These dates identify the archived edition and do not establish historical priority.

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Supporting materials

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I welcome substantive collaboration on independent verification of the every-maximizer estimates and actual-slit continuation, meaningful extensions supported by proof, or development of a journal submission. Authorship and responsibilities would be discussed by mutual agreement and would reflect actual research and writing contributions.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

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Cite this paper

Alec Kriebel. (2026). Nonmonotone Argument and Radial Angle of a Support Slit (Version 1.0, 14 October 2026) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23383961

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Background and prior work