The diagram $(\lambda_1,\mu_1)$

In this paper, we are interested in the possible values taken by the pair $(\lambda_1(\Omega), \mu_1(\Omega))$ the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain $\Omega$. We prove that, without any particular assumption on the class of open sets $\Omega$, 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--Santal\'o diagram and give some conjectures.