Let be distinct real numbers mod 2π andKbe a closed subset of [0,1]. Define Ω to be the unit disk Δ from which the slitsei,aK.j= 1,…n have been removed, and letube the harmonic measure of ∂Δ relative to Ω. Letvbe the corresponding harmonic measure when theaj. are changed toIn case Ω is simply connected Dubinin has proved thatv(0)⩽u(0). It is not known whether this holds in general when Ω is multiply connected. In this paper, we prove it is true when nn⩽ 3. In fact, using*-function arguments, we obtain a much stronger theorem involving integral means.