Operations that Preserve Convexity

Question

If g, f, and q are convex functions, here are three derived functions, which of below would be convex?

• h(x,y)=(f(x)+g(y))^2

• sqrt(f(x)g(y))

• q(f(x))

How to show that they preserves convexity?

I know the basic operations that preserves convexity, but how about those function compositions?

Answer 1

One great resource for this would be Disciplined Convex Programming (DCP).
CVX for MATLAB has a great documentation on the DCP Ruleset.

Given some pre defined atoms it gives a guideline about composition of those atoms to create a convex problem.