Irregular finite order solutions of complex LDE's in unit disc

TL;DR

This paper investigates the growth properties of solutions to complex linear differential equations in the unit disc, establishing conditions for equality of order and lower order, and constructing examples with irregular growth behaviors.

Contribution

It provides a characterization of solutions' growth in relation to the coefficient's properties and introduces new estimates and methods for analyzing lower order growth in complex differential equations.

Findings

Order and lower order are equal iff the coefficient is analytic with specific growth limits.

Constructed examples show solutions with varying lower order despite constant order.

New sharp logarithmic derivative estimates involving lower order are established.

Abstract

It is shown that the order and the lower order of growth are equal for all non-trivial solutions of $f^{(k)}+A f=0$ if and only if the coefficient $A$ is analytic in the unit disc and $\log^+ M(r,A)/\log(1-r)$ tends to a finite limit as $r\to 1^-$. A family of concrete examples is constructed, where the order of solutions remain the same while the lower order may vary on a certain interval depending on the irregular growth of the coefficient. These coefficients emerge as the logarithm of their modulus approximates smooth radial subharmonic functions of prescribed irregular growth on a sufficiently large subset of the unit disc. A result describing the phenomenon behind these highly non-trivial examples is also established. En route to results of general nature, a new sharp logarithmic derivative estimate involving the lower order of growth is discovered. In addition to these estimates, arguments used are based, in particular, on the Wiman-Valiron theory adapted for the lower order, and on a good understanding of the right-derivative of the logarithm of the maximum modulus.