Analysability of finite U-rank types are explored both in general and in the theory ${\rm{DC}}{{\rm{F}}_0}$. The well-known fact that the equation $\delta \left( {{\rm{log}}\,\delta x} \right) = 0$ is analysable in but not almost internal to the constants is generalized to show that $\underbrace {{\rm{log}}\,\delta \cdots {\rm{log}}\,\delta }_nx = 0$ is not analysable in the constants in $\left( {n - 1} \right)$-steps. The notion of a canonical analysis is introduced–-namely an analysis that is of minimal length and interalgebraic with every other analysis of that length. Not every analysable type admits a canonical analysis. Using properties of reductions and coreductions in theories with the canonical base property, it is constructed, for any sequence of positive integers $\left( {{n_1}, \ldots ,{n_\ell }} \right)$, a type in ${\rm{DC}}{{\rm{F}}_0}$ that admits a canonical analysis with the property that the ith step has U-rank ${n_i}$.

New estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f(k)/f(j), where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k>j≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.