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Let (F(X,Y)=sum nolimits _{i=0}^sa_iX^{r_i}Y^{r-r_i}in {mathbb {Z}}[X,Y]) be a form of degree (r=r_sge 3), irreducible over ({mathbb {Q}}) and having at most (s+1) non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality $$begin{aligned} |F(X,Y)|le h end{aligned}$$is (ll s^2h^{2/r}(1+log h^{1/r})). They conjectured that (s^2) may be replaced by s. Let $$begin{aligned} Psi = max _{0le ile s} max left( sum _{w=0}^{i-1} frac{1}{r_i-r_w},sum _{w= i+1}^{s}frac{1}{r_w-r_i}right) . end{aligned}$$Then we show that (s^2) may be replaced by (max  ^3s, se^{Psi })). We also show that if (|a_0|=|a_s|) and (|a_i|le |a_0|) for (1le ile s-1), then (s^2) may be replaced by (slog ^{3/2}s). In particular, this is true if (a_iin {-1,1}).

Contributions to a conjecture of Mueller and Schmidt on Thue inequalities