The diagram $({\lambda }_1,{\mu }_1)$

In this paper, we are interested in the possible values taken by the pair $({\lambda }_1(\mathrm{\Omega }),{\mu }_1(\mathrm{\Omega }))$ the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain $\mathrm{\Omega }$. We prove that, without any particular assumption on the class of open sets $\mathrm{\Omega }$, the two classical inequalities (the Faber-Krahn inequality and the Weinberger inequality) provide a complete system of inequalities. Then we consider the case of convex plane domains for which we give new inequalities for the product ${\lambda }_1{\mu }_1$. We plot the so-called Blaschke–Santaló diagram and give some conjectures.