Let F = GF(q) denote the finite field with q =2nelements. For f(X)∈f[X] we let (formula presented) A deep result of Carlitz and Uchiyama states that if f(X)≠g(X)2+g(X)+b, g(X)∈F[X], b∈F, then │S(f)│≦(deg f-1)q1/2.then This estimate is proved in an elementary way when deg f = 3, 4, 5 or 6. In certain cases the estimate is improved.