Description of growth and oscillation of solutions of complex LDE's
Abstract
It is known that, equally well in the unit disc as in the whole complex plane, the growth of the analytic coefficients $A_0,\dotsc,A_{k2}$ of \begin{equation*} f^{(k)} + A_{k2} f^{(k2)} + \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 nontrivial 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.07934
 Bibcode:
 2019arXiv190507934C
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Complex Variables
 EPrint:
 24 pages. This is a revision of a previously announced preprint. There are many changes throughout the manuscript