It is known that, in the unit disc as well as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc ,A_{k-2}$ of

$$ \begin{align*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geqslant 2, \end{align*} $$

$D(0,R)$

$0<R\leqslant \infty $

We obtain lower bounds for the growth of solutions of higher order linear differential equations, with coefficients analytic in the unit disc of the complex plane, by localising the equations via conformal maps and applying known results for the unit disc. As an example, we study equations in which the coefficients have a certain explicit exponential growth at one point on the boundary of the unit disc and consider the iterated $M$-order of solutions.

This research deals with properties of polynomial regular functions, which were introduced in a recent study concerning Wiman-Valiron theory in the unit disc. The relation of polynomial regular functions to a number of function classes is investigated. Of particular interest is the connection to the growth class Gα, which is closely associated with the theory of linear differential equations with analytic coefficients in the unit disc. If the coefficients are polynomial regular functions, then it turns out that a finite set of real numbers containing all possible maximum modulus orders of solutions can be found. This is in contrast to what is known about the case when the coefficients belong to Gα.

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.