Research Problems in Function Theory — Problem 4.18

Definitions and mathematical statement

For a rational function $f$, its spherical derivative is \[ f^{\#}(z)=\frac{|f'(z)|}{1+|f(z)|^2}, \] with the continuous spherical interpretation at a pole. The degree of $p/q$ in reduced form is $\max(\deg p,\deg q)$; polynomials are included among rational functions. The question concerns degree $N$ and varying positive radii.

First requested bound. Find the least upper bound $\phi(N)$ of the source expression \[ I(r,f)=\frac1r\int_0^r dt\int_{-\pi}^{\pi} \frac{|f'(re^{i\theta})|}{1+|f|^2}\,d\theta, \] for varying $r$ and $f$ of degree $N$. The argument of $f$ in the denominator is omitted; its natural literal reading is also $re^{i\theta}$. With that reading the integrand is independent of $t$, so the expression reduces to $\int_{-\pi}^{\pi}f^{\#}(re^{i\theta})\,d\theta$. It scales by a positive factor $a$ under $f(z)\mapsto f(az)$ and $r\mapsto r/a$, which is incompatible with a finite uniform bound over both $r$ and $f$. Thus a missing argument, factor, normalization, or restriction must be checked before the intended finite extremal problem is determined. Replacing $r$ by $t$ alone would not remove this inverse-length scaling.

The record nevertheless reports \[ \phi(N)=O(N^{1/2}),\qquad \phi(N)\ne O\left((\log N)^{1/2-\varepsilon}\right) \quad(N\to\infty), \] without specifying the quantifier or range for $\varepsilon$. These bounds are retained as claims about the intended source quantity, not asserted for the inconsistent literal reading above.

Further requested bound. The source also asks for a uniform-in-$f$ upper bound for a whole-radius integral involving $f^{\#}$ and angular integration, conjectures it should not be much larger than $\phi(N)$, and says even finiteness was not proved. Its displayed expression has both an outer $dt$ and an additional $dr$ with only radial and angular integration limits, and supplies no argument for $f$. A possible intended functional is \[ J(f)=\int_0^\infty\int_{-\pi}^{\pi}f^{\#}(te^{i\theta})\,d\theta\,dt, \] but this is explanatory and is not silently substituted for the malformed source expression. The phrase "not much larger" gives no precise comparison constant or asymptotic relation. The source especially requests cases $N=2,3,\ldots$ and refers to Littlewood without a complete citation.

The supplied summary records OPEN-TRIAGE, says the item was open in the 2018 edition, and reports no recent resolution located. The transcription problems prevent an unambiguous extremal statement regardless of that status claim.

Short English statement

Determine sharp degree-dependent bounds for radial/angular integrals of spherical derivatives of rational functions, including small degrees. The first printed integral has inconsistent scaling, and the second has mismatched integration variables, so the intended functionals must be recovered before the quoted bounds can be formalized.

Sources

• Local UnsolvedMath registry, permanent ID 2304018, source code catalogue entry, "Research Problems in Function Theory --- Problem 4.18"; supplied status open, classification OPEN-TRIAGE, updated 2026-08-06.
• Hayman, Research Problems in Function Theory (2018), Problem 4.18, https://arxiv.org/abs/1809.07200. The record reports source-TeX extraction and access on 2026-07-29; the 2018 update was retained for status triage. Current status and rights require review; the record describes publicly available arXiv source TeX.
• The source says "See Littlewood" but supplies no complete reference. The asymptotic estimates and the no-recent-resolution-found assessment have no further citations in the record.