Younsi–Ransford monotonicity conjecture for analytic capacity ratios

About 9 years old · traced to

Let ErE_r and FrF_r be the unions of disks determined by points z1,…,znz_1,\dots,z_n and w1,…,wmw_1,\dots,w_m, respectively, with 0<r<δ/20<r<\delta/2, and define

R(r):=γ(Er∪Fr)γ(Er)+γ(Fr).R(r):=\frac{\gamma(E_r\cup F_r)}{\gamma(E_r)+\gamma(F_r)}.

Younsi–Ransford conjecture. For all z1,…,zn,w1,…,wmz_1,\dots,z_n,w_1,\dots,w_m, the function R(r)R(r) is decreasing on 0<r<δ/20<r<\delta/2.

Numerical experiments suggest this monotonicity, alongside the conjectured subadditivity of analytic capacity, but no resolution is given here.

References

Primary source

Malik Younsi, “Analytic Capacity : computation and related problems”, arXiv:1705.02912 (2017).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.