AUSTRIAN-POLISH 2004 Problem 2

In a triangle $ABC$ let $D$ be the intersection of the angle bisector of $\gamma$ , angle at $C$ , with the side $AB.$ And let $F$ be the area of the triangle $ABC.$ Prove the following inequality:

$$2 \cdot \ F \cdot \left( \frac{1}{AD} -\frac{1}{BD} \right) \leq AB.$$