On the Hersch–Payne–Schiffer inequalities for Steklov eigenvalues

We prove that the Hersch–Payne–Schiffer isoperimetric inequality for the $n$th nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all $n\ge 1$. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of $n$ identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch–Payne–Schiffer inequality for $n=2$ and show that it is strict in this case.