  This paper is devoted to a comparison between the normalized first
(non-trivial) Neumann eigenvalue $|\Omega| \mu_1(\Omega)$ for a Lipschitz open
set $\Omega$ in the plane, and the normalized first (non-trivial) Steklov
eigenvalue $P(\Omega) \sigma_1(\Omega)$. More precisely, we study the ratio
$F(\Omega):=|\Omega| \mu_1(\Omega)/P(\Omega) \sigma_1(\Omega)$. We prove that
this ratio can take arbitrarily small or large values if we do not put any
restriction on the class of sets $\Omega$. Then we restrict ourselves to the
class of plane convex domains for which we get explicit bounds. We also study
the case of thin convex domains for which we give more precise bounds. The
paper finishes with the plot of the corresponding Blaschke-Santal\'o diagrams
$(x,y)=\left(|\Omega| \mu_1(\Omega), P(\Omega) \sigma_1(\Omega) \right)$.
