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 $$\max (T_G(2,0),T_G(0,2))⩾T_G(1,1)\mathrm{.}$$ Later Jackson proved that for any matroid $M$ without loops and coloops we have $$T_M(3,0)T_M(0,3)⩾T_M(1,1{)}^2\mathrm{.}$$ The value $3$ in this statement was improved to $2.9243$ by Beke, Csáji, Csikvári and Pituk. In this paper, we further improve on this result by showing that $$T_M(2.355,0)T_M(0,2.355)⩾T_M(1,1{)}^2\mathrm{.}$$ We also prove that the Merino–Welsh conjecture is true for matroids $M$, where all circuits of $M$ and its dual $M^{\ast }$ have length between $\mathrm{\ell }$ and $(\mathrm{\ell }-2{)}^2({\mathrm{\ell }}^2-4\mathrm{\ell }+2)$ for some $\mathrm{\ell }⩾4$.