On functions satisfying certain differential inequalities

Abstract

then f(x) = ce-x. In ?2 a new proof is given using Liouville's theorem. So far as I am aware, the method of proof does not apply to any essentially more general problems, depending as it does on unique properties of the analytic function ez. However, I have obtained a few similar results by other methods, and to establish a sufficiently broad framework to contain all cases, it is convenient to pose a somewhat indefinite general problem: PROBLEM A. Let C1 be a linear space of functions; let T be an operator on Ci to C1; and let .| be a norm defined on C1. When does fEC1 and II TkfII _1, k=O, 1, 2, ,imply that fC C2 where C2 is a finite-dimensional subspace of C1? A narrower framework will also be considered: PROBLEM B. Let T be a differential operator and let g be a function such that Tg -g (or +g). When does I Tkf| <gi, k=O, 1, 2, imply f = cg? A very simple result of type A is obtained in ?3 by taking T to be a self-adjoint operator on a Hilbert space. Differential operators are natural choices for T, and some special cases are spelled out using Fourier and Hermite expansions. A corollary of type B is given. In ?4, I consider the possibility of generalizing results for particular operators T to results for conjugate operators of the form STS-1. For Problem B a natural choice for S is an operator which acts in effect to change the variables of the differential equation Tf = -f. By employing this device, results are obtained for example for T=pl(x)D +po(x) and for T=AD2+BD+C using respectively Tagamlitzki's theorem and a Fourier expansion.