Sharper estimates for analytic functions in two dimensions

Investigate whether, in dimension k = 2 with R^2 identified with the complex plane, the growth-transfer estimates for subharmonic functions u(z) = log|f(z)|, where f is analytic on Ω \ B, can be improved relative to those established for general subharmonic functions under comparable hypotheses on A and the majorant g.

Background

The paper derives corollaries for analytic functions by applying the subharmonic results to u = log|f|, yielding bounds in terms of dist(z,B).

Given the special structure of analytic functions, the authors ask whether one can obtain sharper bounds than those available for general subharmonic functions in the planar case.

References

We end the article with the following open questions. Suppose that k=2, and identify R2 with the complex plane. Are the estimates for subharmonic functions of the form u(z)=\log |f(z)|, where f is analytic on \Omega \ B, better than those for general subharmonic functions?

— Self-improving estimates of growth of subharmonic and analytic functions (2508.04496 - Bello et al., 6 Aug 2025) in Question 5, end of Section 5 (Application of quantitative Domar's results to our problem)