Is a function of admissible heuristics in A* search admissible?

Question

I don’t understand how to approach this problem.

$h_1, h_2, h_3$ are three admissible heuristics for an optimisation problem to be solved using A* search. Is the heuristic defined by

$$h(n) = \frac{\sqrt{h_1(n)h_2(n)}+ 2h_3(n)}{3} $$

for any node n of the search graph, admissible?

This is all the information I have, I have no problem specifics or anything else. Any help would be great! Thanks

Answer 1

Let $f(n)$ be the optimal cost to node $n$. Suppose $h_1(n) \ldots h_3(n)$ are positive lower bounds for $f(n)$, i.e. admissible heuristics. Can you prove that $$\frac{\sqrt{h_1(n) h_2(n)} + 2 h_3(n)}{3}$$ is no more than $f(n)$? You might start by proving that $\sqrt{h_1(n) h_2(n)} \le f(n)$.