  Among all $C^{\infty}$ bounded domains with equal volume, we show that the
second eigenvalue of the Robin plate is uniquely maximized by an open ball, so
long as the Robin parameter lies within a particular range of negative values.
Our methodology combines recent techniques introduced by Freitas and Laugesen
to study the second eigenvalue of the Robin membrane problem and techniques
employed by Chasman to study the free plate problem. In particular, we choose
eigenfunctions of the ball as trial functions in the Rayleigh quotient for a
general domain; such eigenfunctions are comprised of ultraspherical Bessel and
modified Bessel functions. Much of our work hinges on developing an
understanding of delicate properties of these special functions, which may be
of independent interest.
