Zero separation results for solutions of second order linear differential equations

Abstract

The oscillation of solutions of f″+Af=0 is discussed by focusing on four separate situations. In the complex case A is assumed to be either analytic in the unit disc D or entire, while in the real case A is continuous either on (−1,1) or on (0,∞). In all situations A is expected to grow beyond bounds that ensure finite oscillation for all (non-trivial) solutions, and the separation between distinct zeros of solutions is considered.In the complex case, it is shown that the growth of the maximum modulus of A determines the minimal separation of zeros of all solutions, and vice versa. This gives rise to new concepts called zero separation exponents, which measure the separation of zeros of either all solutions or of individual analytic functions. In D these quantities are defined in terms of the hyperbolic distance, while in the complex plane the Euclidean distance is used. As a by-product of these findings, the 1955-result of B. Schwarz, which asserts that supz∈D|A(z)|(1−|z|2)2<∞ if and only if the zero-sequences of all solutions are separated in the hyperbolic sense, is rediscovered. The striking plane analogue established reveals that the Euclidean distance between all distinct zeros of every solution is uniformly bounded away from zero if and only if A is a constant. As an outgrowth of the results, new information on the zero distribution of solutions in the classical polynomial coefficient case is also obtained. The main results are proved by using a method of localization, which naturally induces characterizations of certain subclasses of locally univalent functions in terms of the growth of their pre-Schwarzian and Schwarzian derivatives.In the real case, it is shown that the separation of zeros of non-trivial solutions is restricted according to the growth of A, but not conversely.