Modulus estimate with intersecting annuli (quasi-additivity)

In general for annulus $$A\subset \mathbb{C}$$ if $$A_{1},A_{2}....\subset A$$ are disjoint annuli inside it, then we have $$mod(A)=\frac{1}{2\pi}\int_{A}\int_{A} \frac{1}{|z|^{2}}dz>\frac{1}{2\pi}\int_{\cup A_{i}}\int_{\cup A_{i}} \frac{1}{|z|^{2}}dz\geq \sum mod(A_{i}).$$

In my case I have

• shrinking annuli $$A_{i}$$ separating zero from infinity
• that could possibly intersect on some places $$Q_{i,j}:=A_{i}\cap A_{j}\neq \varnothing$$
• with, hopefully, controlled diameters $$diam(Q_{i,j})\leq d_{i,j}$$ going to zero as the annuli shrink
• and the sum $$\sum_{i=1}^{N} mod(A_{i})>c_{0}N$$ grows linearly.

Let $$U_{N}$$ be an annulus that contains $$\bigcup_{i=1}^{N} A_{i}$$.

Q: I am hoping to get a growth for $$mod(U_{N})$$ eg. $$mod(U_{N})\geq c_{1}\sum_{i=1}^{N} mod(A_{i}).$$

Thank you for any suggestions.

One interesting paper in this direction is "The Quasi-Additivity Law in Conformal Geometry". I will state their result in full generality for the interested reader.

Let S stand for a compact Riemann surface with boundary then for compact subset $$K\subset S$$ let $$W(S,K)=\frac{1}{L(S,K)}$$ be the extremal width (reciprocal of extremal length) of paths connecting $$\partial S$$ to $$K$$ inside $$S\setminus K$$.

Consider open sets $$A_{i}\subset S$$ for i=1,..,N , whose closure is a Riemann surface of finite type (not necessarily connected) with smooth boundary.

Let $$X:=W(S,\cup_{i=1}^{N} A_{i})$$, $$Y:=\sum_{i=1}^{N}W(S,A_{i})$$ and $$Z:=\sum_{i=1}^{N}W(S\setminus \cup_{j\neq i}^{N} A_{j},A_{i})$$.

If $$Y<\xi Z$$ for some $$\xi\geq 1$$, there exists constant $$K$$ depending on $$\xi$$ and the betti numbers of $$S\setminus \cup_{j\in M} A_{j}$$ for $$M\subset \{1,...,N\}$$, such that $$Y\geq K\Rightarrow Y\leq 2\xi X.$$