The general answer is to use Hooke's Law: F = k*x

http://www.dummies.com/education/science/physics/how-to-calculate-a-spring-constant-using-hookes-law/

Let's start by assuming that the entire plastic part is a spring. Meaning that all of it will deform to some extent, when that force is applied. As opposed to only the curved part deforming, and the straight parts remaining undeformed. To derive what F must be, you need to know x (2 mm) and the "spring constant" k.

But you have too many unknowns. The force required will vary in direct proportion to the spring constant k.

A slightly more complicated variant would be if only the curved part is a spring, with an associated k, in which case you would have a spring and a lever problem. The force necessary would be a function of k, as before, and also the length of the straight lever arm. The force required would be reduced as the lever arm length is increased. For this, you also would need to know the force required at some point close to the curved part, and then define the lever arm length, to see how that force is reduced at the point where you measure the 2 mm deflection.

Hi

You need the following information:

- all dimensions     - elastic modulus       - please mark all the fixing points/lengths/parts

Then you can use the singularity function for each part separately and superimpose the results.  

Is the lower part made of the same material as the upper part? How they are actually connected on the right side? What is the function of the extra part there? Where are the actual support points and how these should be modelled? If these are clarified, then you may use an advanced FEM package able to handle constrained deformation. Sub-structuring might also be useful, but may not be needed, apart from for a comparison of the two-way computed results. 

This seems well suited for finite element analysis. Model the part, fix the boundaries which don't move; force a deflection of 2mm over a series of smaller substeps on the end of the movable part.