Research Problems in Function Theory — Problem 6.89

Definitions and mathematical statement

The usual normalized class $S^*(1/2)$ consists of univalent holomorphic functions $f$ on $\mathbb D$ with $f(0)=0$, $f'(0)=1$, and \[ \operatorname{Re}\frac{zf'(z)}{f(z)}>\frac12, \] using the removable value at $0$. The supplied text gives only analyticity and the displayed inequality, without these normalizations or a zero convention, so the exact intended class must be checked.

For $f$ in this class, $0<r<1$, and $0\leq\theta<2\pi$, R. R. Hall's first question is printed as \[ \frac{\int_0^r|f'(te^{i\theta})|\,dt}{|f(re^{-i\theta})|} \leq\frac{\arcsin r}{r}. \] It simultaneously describes the left side as image-radius length divided by the distance between that image's endpoints. Those descriptions conflict: the image of the radius $te^{i\theta}$ has chord length $|f(re^{i\theta})-f(0)|$, which under normalization is $|f(re^{i\theta})|$, with the opposite sign from the printed denominator. Thus the natural geometric candidate would instead be \[ \frac{\int_0^r|f'(te^{i\theta})|\,dt}{|f(re^{i\theta})|} \leq\frac{\arcsin r}{r}, \] but this correction is not verified by the record. The source says the inequality holds for $f(z)=z/(1-z)$ and that the left side is always at most $\pi/2$. These are retained as reported claims whose precise sign convention must be checked; the example has reflection-symmetric moduli and does not identify which sign was intended. Here $\arcsin r$ is the real inverse sine on $(0,1)$.

The second question asks about \[ \frac{\int_{z_1}^{z_2}|f'(t)|\,|dt|}{|f(z_1)-f(z_2)|} \] for the smaller class of convex functions. In the standard normalized interpretation, convex means $f$ is univalent and $f(\mathbb D)$ is convex. Distinct endpoints $z_1\ne z_2$ are required. The integration path is not stated; if it is the straight segment, the numerator is \[ |z_2-z_1|\int_0^1|f'(z_1+s(z_2-z_1))|\,ds, \] the length of that segment's image. The record supplies no precise bound or optimization target for this second request and no path convention. The supplied summary reports openness as of 2018 and no definitive resolution located.

Short English statement

Hall asks for a sharp bound comparing image arc length with its chord for functions starlike of order one half, and an analogue for convex functions. The printed first denominator uses the reflected radius, contrary to the geometric explanation; the second integral has no stated path. These ambiguities require source review.

Sources

• Local UnsolvedMath registry, permanent ID 2306089, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 6.89"; status open, classification OPEN-TRIAGE.
• Hayman, Research Problems in Function Theory (2018), Problem 6.89, attributed to R. R. Hall, https://arxiv.org/abs/1809.07200. The record identifies extraction from public arXiv source TeX, accessed July 29, 2026; the 2018 update was retained for status triage.
• The source reports the example $z/(1-z)$ and universal $\pi/2$ bound without a separate citation. Its source label J6.89 refers to the displayed inequality but does not resolve the reflected-angle discrepancy.
• The supplied research summary reports openness as of the 2018 edition and no definitive resolution located, with no additional authoritative reference or precise search date.