Littlewood constants $\alpha$ and $\beta$

Definitions and mathematical statement

Let $D=\{z\in{\mathbb{C}}:|z|<1\}$. With $m$ interpreted as planar area measure, define, for integers $n\geq1$, \[ \phi(n)=\sup_{\substack{p\in{\mathbb{C}}[z]\\\deg p=n}} \int_D\frac{|p'(z)|}{1+|p(z)|^2}\,\mathrm{d}m(z), \qquad \alpha=\limsup_{n\to\infty}\frac{\log\phi(n)}{\log n}. \] Only $n\geq2$ enters the limiting quotient. Multiplying area measure by a fixed positive constant does not alter this exponent. The source imposes no monicity, coefficient, or boundary normalization on the degree-$n$ polynomials.

For a regular compact set $E$ in the source's potential-theoretic sense, the record refers to a constant $\beta_E$ defined by the growth of lengths of level curves of its Green function, and then sets \[ \beta=\sup\{\beta_E:E\text{ is connected and is an admissible regular compact set}\}. \] The question asks for a connection between these analytic and geometric exponents, in particular whether $\alpha=\beta$.

The source does not give the formula for $\beta_E$, the Green-function normalization and domain, the limiting level parameter, the length convention, or the admissibility conditions on $E$. Thus the last two displays retain the source's named quantity but do not define a self-contained geometric exponent. These missing data must be recovered before the proposed equality can be treated as a precise independent statement.

Short English statement

Compare the polynomial spherical-derivative integral growth exponent $\alpha$ with the maximal Green-level-curve length exponent $\beta$ for connected compact sets. Is $\alpha=\beta$? The supplied record omits the definition of the geometric exponent.

Sources

• Local UnsolvedMath registry, permanent ID 3700029, exact source code catalogue entry, Littlewood constants, Question A; proposed by Alexandre Eremenko and cited proposers, proposed year 2002. Registry status partially solved, classification PARTIAL-PROGRESS.
• Alexandre Eremenko, My favorite unsolved problems, https://www.math.purdue.edu/~eremenko/uns1.html. Supplied source item: "Littlewood constants, Question A"; extraction pdf-text, accessed July 29, 2026. The research summary names Eremenko's littlewood.pdf but supplies no separate full citation or URL for it. No PDF was accessed here.
• The supplied research summary describes established connections and bounds, and says the equality remains unresolved. It reports $\beta\leq1/4$ and $\alpha\leq3+2\sqrt2$, attributing results generally to Eremenko and predecessors without a precise theorem or reference. These numerical claims and their normalization require verification; they are not used to complete the missing definition of $\beta_E$. Its description of $\phi(n)$ as growing "roughly like $n^\alpha$" is not substituted for the exact limsup definition.
• Provenance states the record was retained after the author-maintained index's explicit solved/counterexample updates through the access date. Current status and rights are flagged NEEDS_REVIEW: public access was verified, but redistribution permission was not established.