I've yet to see realistic examples of improvements, at least for domains I work in. Those pushing FP either invent contrived "lab toy" circumstances where FP shines, use vague buzzwords that don't describe any concrete benefit, or merely expose the weakness of a particular language or API design (such as Java's poor OOP model).
If you want to convince people, show them realistic examples and explain the benefits in concrete and measurable terms. Is that asking too much?
They may say, "If you use it long enough, you'll finally get it"; but there are so many tools and ideas competing for our time that I'll have to ask for more up-front evidence, if you don't mind.