Dennis Stowe

A generalization of Nehari's univalence criterion.

This note is a sequel to our paper [OS] where we generalized the Schwarzian derivative to conformal mappings of Riemannian manifolds. There we found that many of the phenomena familiar from the classical theory have counterparts in the more general setting. Here we advance this another step by giving a generalization of the well known univalence criterion of Nehari [N]. Despite its relatively advanced age, this result continues to generate interest, see [L]. The argument used here in the general case, if specialized to the situation considered by Nehari, gives a somewhat different and a more geometric proof of his theorem than is often presented. We want to keep this note short, since the proof of the Theorem is really quite simple, and also fairly self-contained. We shall need a number of facts from our earlier paper and we collect them here with very little additional discussion. We refer the reader to that paper for more details.

OSCILLATION OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

In this note we study the zeros of solutions of differential equations of the formu !! +pu=0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.

Functions with prescribed quasisymmetry quotients

Let f be a continuous, increasing function of R into itself. Let k f (x, h) = f(x + h) − f(x) f(x) − f(x − h) , x ∈ R, h > 0. Observe that if g = af + b, a, b ∈ R, then k g = k f. One can show conversely that if k g = k f , then g = f up to a real affine transformation. The quantity k f is called the quasisymmetry quotient of f. A function is c-quasisymmetric, c ≥ 1, if 1 c ≤ k f (x, h) ≤ c. (1) The condition (1) also implies that f (±∞) = ±∞. We refer to the book by Lehto and Virtanen [1] for a thorough discussion of the role of quasisymmetric functions in the theory of quasiconformal mappings. Due to its simple form the quasisymmetry quotient necessarily satisfies any number of algebraic identities. Two of these are sufficient, as we shall prove in the following existence theorem. If in addition k(x, h) is bounded between 1/c and c, then of course f is c-quasisymmetric.