Some inequalities for the Tutte polynomial

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We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*. For this latter class we give a proof that Tₘ(a,a) <= max Tₘ(2a,0), Tₘ(0,2a) for a >= 2. We conjecture that 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.