The functional form of Mahler conjecture for even log-concave functions in dimension $2

Let $\varphi: \mathbb{R}^n\to \mathbb{R}\cup\{+\infty\}$ be an even convex function and $\mathcal{L}{\varphi}$ be its Legendre transform. We prove the functional form of Mahler conjecture concerning the functional volume product $P(\varphi)=\int e^{-\varphi}\int e^{-\mathcal{L}\varphi}$ in dimension 2: we give the sharp lower bound of this quantity and characterize the equality case.The proof uses the computation of the derivative in $t$ of $P(t\varphi)$ and ideas due to Meyer for unconditional convex bodies, adapted to the functional case by Fradelizi-Meyer and extended for symmetric convex bodies in dimension 3 by Iriyeh-Shibata.