The quantities $a_i$ are all positive real numbers.
(a) Show that
$$a_1.a_2 \leq \left(\frac{a_1 + a_2}{2}\right)^2$$
(b) Hence prove, by induction on m, that
$$a_1.a_2.......a_p \leq \left(\frac{a_1+a_2+......+a_p}{p}\right)^p$$ ,
where $p = 2^m$ with m a positive integer. Note that each increase of m by unity doubles the number of factors in the product.
The (a) part of the problem can be proved by using the expansion of $(a_1+a_2)^2$, for the (b) part: taking $m=1$ gives $p = 2$ (base case), which is same as the (a) part.
Assuming that: $$a_1.a_2.....a_P \leq \left(\frac{a_1+a_2+.......+a_P}{P}\right)^P$$ (taking $m=M$ so that $p = P = 2^M$);
Substituting $m = M+1 \Rightarrow p = 2P = 2^{M+1}$ on the L.H.S. : $$a_1.a_2....a_P.a_{P+1}.a_{P+2}.....a_{2P} \Rightarrow$$ $$\left(\frac{a_1+a_2+.....+a_P}{P}\right)^P.a_{P+1}.a_{P+2}.....a_{2P}$$ How to proceed from here on?