I would like to find a way to find the most optimale constant Poincaré's inequality in a first time and Gagliardo-Niremberg inequality in a second time but in very simple case, i.e. in dimension $1$, for radial functions, and maybe in the first time for $u\in W^{1,1}(\mathbb R)$ (or in fact $\mathcal C^1(\mathbb R)$ if it's easier). I found many articles on the internet as
1) Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions
2) BEST CONSTANT IN A THREE-PARAMETER POINCARE´ INEQUALITY
3) https://hal.archives-ouvertes.fr/hal-01318727/document
4) Best Constant in Sobolev Inequality
But it's for the moment much to complicate for me. I really want to find easy cases for the moment before to go in generalization.