Find equation for a function that converges below identity function like the picture

Question

I'm trying to come up with a function that behaves like this:

Basically, the function is below the identity function x=y, and some upper bound (here, it is 0.25). And there is a parameter that controls how close it is to the blue lines (x=y and upper bound). This seems simple but I can't figure it out for the life of me.