Research Problems in Function Theory — Problem 6.74

Definitions and mathematical statement

Let $f$ be injective and analytic on $\mathbb D$, normalized by \[ f(z)=z+\sum_{k=2}^{\infty}a_kz^k, \qquad |f(z)|\leq M\quad(z\in\mathbb D). \] For an integer $n\geq1$, its $n$th Taylor section is \[ s_n(z)=z+\sum_{k=2}^{n}a_kz^k, \] with an empty sum when $n=1$. Since $s_n$ is a polynomial, \[ \|s_n\|_{\mathbb T}=\max_{0\leq t\leq2\pi}|s_n(e^{it})| \] is defined without requiring boundary values of $f$.

The source prints the requested quantity as $\sup_t\max_{0\leq t\leq2\pi}|s_n(e^{it})|$. The inner maximum already binds $t$, so the outer subscript does not specify a separate optimization variable. For a fixed $f$, the inner maximum is the circle norm above. If the intended task is the sharp bound over all normalized functions with bound $M$, its precise form is \[ A_n(M)=\sup\left\{\|s_n\|_{\mathbb T}: f\text{ normalized univalent in }\mathbb D, \ \sup_{z\in\mathbb D}|f(z)|\leq M\right\}, \] for parameters with a nonempty class. F. Holland's problem is to determine the source's intended extremal value; the supremum-over-$f$ interpretation and whether an exact value or an asymptotic in $n$ is requested must be confirmed rather than silently repairing the printed subscript.

Short English statement

Determine the sharp unit-circle size of Taylor sections of normalized bounded univalent functions. The displayed outer supremum uses the same variable as the inner maximum and therefore leaves the optimization class unclear. The supplied triage reports the problem open in Hayman's 2018 update, mentions related partial-sum work by Pommerenke and others without identifying a theorem, and says no definitive resolution was located. No independent status review is claimed.

Sources

• UnsolvedMath permanent ID 2306074, source code catalogue entry, "Research Problems in Function Theory --- Problem 6.74." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018 source update), Problem 6.74, attributed to F. Holland: https://arxiv.org/abs/1809.07200. Supplied source-TeX extraction accessed 2026-07-29. The metadata retains the update for status triage and marks status/release and rights review NEEDS_REVIEW.
• The supplied summary mentions Pommerenke and unspecified others for related partial-sum problems, without exact statements or references. No specific resolution is inferred.