An inequality for double tangents

For a regular closed curve on the plane it is known that E = I + X + 2 F where E, I, X and F are the numbers of external double tangents, internal double tangents, self-intersections, and inflexion points respectively. It is proven here that if F = 0 then I is even and I < (2X + 1)(X 1). Furthermore, examples are given which show that if the four tuplet (E, I, X, F) of nonnegative integers satisfies (a) F even, (b) E = I + X + F, and (c) if F = 0 then I is even and I < X(X 1), then there is a regular closed plane curve which realizes these values.