Proving convexity of a function in general can be a bit of a pain, but there are few tricks.
1. If a function is quadratic, check if the Hessian is positive semidefinite. In that case, the function is convex. For a general function, you technically check this condition at every point, but I typically find that difficult.
2. Frankly, the best way to prove convexity is to see if you can do so constructively. This doesn't always work, but when it does, it's the easiest. Note, let f and g be convex and f be monotonically increasing. Then, is convex. In order to see this, notice that
where we use convexity in both inequalities and monotonicity of f is the first inequality.
Now, as to your specific functions:
1. : No. It's easiest to see in one dimension. is convex. However, is not since , , but . According to convexity, we need . Or, simply graph the function and you'll see a bump at the origin.
2. : No. Let . Then, we have l , which is actually concave. It's easiest just to graph, but you can verify this by looking at the second derivative of .
3. Yes. is convex and monotonically increasing. is convex. From the composition rule above, the result is convex.