Question: if f is analytical and $ |f(z)| < M $ for $ |z| =< R $ find an upper bound for |f('n)(z)| in $ |z| =< \rho < R $ (where f('n) means the nth derivative of $f$).
So I cited Cauchy's Inequality, to find the bound $\frac{n!M_R} {R^n}$ , but since we have the additional limit of $\rho$ I think we should be able to find a better bound. The geometric interpretation is that the the circle of radius $\rho$ limits the radius of $|z|$. Is there further geometric intuition that I should see to solve this problem?