Better estimates for Littlewood constants

Definitions and mathematical statement

For a polynomial $p$, put $p^\#(z)=|p'(z)|/(1+|p(z)|^2)$, and let $dA$ denote Euclidean area. Define \[ \Phi(N)=\sup_{\deg p=N}\int_{|z|<1}p^\#\,dA, \qquad \alpha=\limsup_{N\to\infty}\frac{\log\Phi(N)}{\log N}. \] For a connected compact plane set $E$ of positive logarithmic capacity, assume its Green function $G_E$ with pole at infinity tends continuously to zero at the boundary of the unbounded complementary component. Extend it by zero over the remaining components. Set \[ \ell_E(\varepsilon)=\mathcal H^1\{G_E=\varepsilon\}, \quad \beta_E=\limsup_{\varepsilon\downarrow0} \frac{\log\ell_E(\varepsilon)}{-\log\varepsilon}, \quad \beta=\sup_E\beta_E . \] Here $\mathcal H^1$ is ordinary curve length.

A parameter $c$ is hyperbolic if the critical orbit of $p_c(z)=z^2+c$ tends to an attracting cycle, possibly infinity. On its Julia set choose $w$ and define the pressure \[ P_c=\lim_{m\to\infty}\frac1m \log\sum_{p_c^m(z)=w}|(p_c^m)'(z)|^{-1}, \qquad P_*=\sup_{c\ \mathrm{hyperbolic}}P_c. \] For this expanding dynamics the pressure is independent of the Julia-set base point.

Obtain sharp, rigorously justified bounds for $\alpha,\beta,P_*$, with the goal of determining their values. Computer-assisted estimates are admissible if numerical and truncation errors are controlled. Bounds for a single polynomial, compact set, or parameter provide examples, while universal upper bounds must cover the full indicated class. The definitions fix the normalizations rather than treating historical numerical estimates as exact constants.

Short English statement

Improve rigorous bounds for the polynomial spherical-derivative exponent, the Green-level length exponent, and maximal quadratic pressure.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700030 (AMR-036-0030), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Littlewood constants, Question B. Original source indexed by AMR-036-0030. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Some constants coming from the work of Littlewood, Question B. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/lit.pdf.