How to interpret the general term of this inequality,

Question

The assumption is that $$M(2R) \le 2^N M(R)$$

Then it follows that

$$M(R) \le 2^N M(R/2)$$

$$\le 2^N 2^N M(R/4)$$ $$=2^{2N}M(R/2^2)$$ $$.$$ $$.$$ $$.$$ $$\le2^{Nlog_2(R)}M(\alpha)$$ $$=R^N M(\alpha)$$ for $0<\alpha\le1$.

I don't know how the solution that I am studying got to the last inequality and subsequently the equality $R^N M(\alpha)$.

Thanks,

EDIT: To add a little more context, M(R) is the maximum of a complex function f(z), where the max is taken over |z|=R. So, I think the goal is to find conditions on this M(R) to show that it is finite and achieved, when R grows to infinity.

Answer 1

How many times do you have to divide $R$ by $2$ until you get a number $\le1$? That is, what is the smallest $k$ such that $$ \frac{R}{2^k}\le1? $$