# Description of growth and oscillation of solutions of complex LDE's

## TL;DR

This paper explores how the growth of coefficients in complex linear differential equations influences the solutions' growth and oscillation, providing a unified approach that extends beyond classical finite order cases.

## Contribution

It introduces a flexible, unified framework for analyzing growth and oscillation of solutions in complex linear differential equations using new measures and estimates.

## Key findings

- Provides a general growth-oscillation relationship for solutions
- Extends analysis to cases beyond finite order growth
- Uses new integrated estimates for logarithmic derivatives

## Abstract

It is known that, equally well in the unit disc as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc,A_{k-2}$ of \begin{equation*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geq 2, \end{equation*} determines, under certain growth restrictions, not only the growth but also the oscillation of its non-trivial solutions, and vice versa. A uniform treatment of this principle is given in the disc $D(0,R)$, $0<R\leq \infty$, by using several measures for growth that are more flexible than those in the existing literature, and therefore permit more detailed analysis. In particular, results obtained are not restricted to cases where solutions are of finite (iterated) order of growth in the classical sense. The new findings are based on an accurate integrated estimate for logarithmic derivatives of meromorphic functions, which preserves generality in terms of three free parameters.