Around the Merino--Welsh conjecture: improving Jackson's inequality

The Merino-Welsh conjecture states that for a graph $G$ without loops and bridges the Tutte polynomial $T_G(x,y)$ satisfies the inequality $$\max(T_G(2,0),T_G(0,2))\geqslant T_G(1,1).$$ Later Jackson proved that for any matroid $M$ without loops and coloops we have $$T_M(3,0)T_M(0,3)\geqslant T_M(1,1)^2.$$ The value $3$ in this statement was improved to $2.9243$ by Beke, Cs\'aji, Csikv\'ari and Pituk. In this paper, we further improve on this result by showing that $$T_M(2.355,0)T_M(0,2.355)\geqslant T_M(1,1)^2.$$ We also prove that the Merino--Welsh conjecture is true for matroids $M$, where all circuits of $M$ and its dual $M^*$ have length between $\ell$ and $(\ell-2)^2(\ell^2-4\ell+2)$ for some $\ell\geqslant 4$.