DESCRIPTION OF GROWTH AND OSCILLATION OF SOLUTIONS OF COMPLEX LDE’S

Abstract

Abstract 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 $$ \begin{align*} f^{(k)} + A_{k-2} f^{(k-2)} + \dotsb + A_1 f'+ A_0 f = 0, \quad k\geqslant 2, \end{align*} $$ determines, under certain growth restrictions, not only the growth but also the oscillation of the equation’s nontrivial solutions, and vice versa. A uniform treatment of this principle is given in the disc $D(0,R)$ , $0<R\leqslant \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, the results obtained are not restricted to cases where the 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.