Decay of separated subharmonic level components

Definitions and mathematical statement

Let $A=\{z:1<|z|<2\}$ and let $u$ be subharmonic there, not identically $-\infty$. Suppose that $D_1,D_2,\ldots$ are pairwise disjoint open subsets of $A$, each a union of connected components of $\{u<0\}$. For each $k$, choose a number $\delta_k>0$ such that for every radius $r\in(1,2)$, \[ \int_{\{\theta\in[0,2\pi):\,re^{i\theta}\in D_k\}} u(re^{i\theta})\,d\theta \leq-\delta_k. \] Thus each group of negative components contributes a fixed positive amount of negative angular integral on every circle of the annulus. The lower amount may depend on $k$, but not on $r$.

Determine the possible decay of the positive sequence $(\delta_k)$ under these hypotheses. The substantive issue is sharp necessary restrictions and matching constructions, up to reordering the components and multiplying $u$ and all $\delta_k$ by a common positive constant. In particular, determine the sharp summability consequences: for which exponents $p>0$ must every admissible sequence satisfy $\sum_k\delta_k^p<\infty$, and what occurs at the limiting exponent?

The source gives exponentially decaying examples and a universal power-summability restriction, leaving a gap between realizations and upper restrictions. The target is to close that gap for general subharmonic functions. No harmonicity, finite number of components, or radial symmetry assumption is imposed.

Short English statement

How slowly can the uniformly negative contributions of disjoint subharmonic level components decay?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700022 (AMR-036-0022), 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, Potential theory, Problem 3. Original source indexed by AMR-036-0022. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Problems of potential theory arising in value distribution, Problem 3. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/potvd.pdf.