Some inequalities for the Tutte polynomial

We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portion of the line x+y=px+y=p lying in the positive quadrant. Every coloopless paving matroid is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases. For this latter class we give a proof that TM(a,a)≤max{TM(2a,0),TM(0,2a)}TM(a,a)≤max{TM(2a,0),TM(0,2a)} for a≥2a≥2. We conjecture that TM(1,1)≤max{TM(2,0),TM(0,2)}TM(1,1)≤max{TM(2,0),TM(0,2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.