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