A sharp estimate of area for sublevel-set of Blaschke products

Let D be the unit disk in the complex plane. Among other results, we prove the following curious result for a finite Blaschke product: B(z)=e ⁱˢ∏ₖ₌₁ᵈ z-aₖ1-z aₖ̄. The Lebesgue measure of the sublevel set of B satisfies the following sharp inequality for t ∈ [0,1]: |∈ D:|B(z)|<t\|≤ π t2/d, with equality at a single point t∈(0,1) if and only if aₖ=0 for every k. In that case the equality is attained for every t.