  Let $\Phi$ : R n $\rightarrow$ R $\cup$ {+$\infty$} be an even convex
function and L$\Phi$ be its Legendre transform. We prove the functional form of
Mahler conjecture concerning the functional volume product P ($\Phi$) = e
--$\Phi$ e --L$\Phi$ 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$\Phi$) and ideas due to Meyer [M] for unconditional
convex bodies, adapted to the functional case by Fradelizi-Meyer [FM2] and
extended for symmetric convex bodies in dimension 3 by Iriyeh-Shibata [IS] (see
also [FHMRZ]).
