Valence and Oscillation of Functions in the Unit Disk

We investigate the number of times that nontrivial solutions of equations u′′ + p(z)u = 0 in the unit disk can vanish—or, equivalently, the number of times that solutions of S(f) = 2p(z) can attain their values—given a restriction |p(z)| ≤ b(|z|). We establish a bound for that number when b satisfies a Nehari-type condition, identify perturbations of the condition that allow the number to be infinite, and compare those results with their analogs for real equations φ′′ + q(t)φ = 0 in (−1, 1). This paper investigates the number of times that nontrivial solutions of an equation u′′ + p(z)u = 0 in the unit disk D ⊆ C can vanish. Which conditions |p(z)| ≤ b(|z|) imply that the number of zeroes is finite? In terms of b, how many zeroes can there be? And how do the answers to those questions compare with what happens with equations φ′′ + q(t)φ = 0 for real-valued functions in (−1, 1)? The results for the complex setting are equivalent to statements about the valence of a locally injective, meromorphic mapping f in D whose Schwarzian derivative S(f) = (f ′′/f ′)′ − 1 2 (f ′′/f ′)2 satisfies a bound |Sf(z)| ≤ 2b(|z|). Because every solution of S(f) = 2p is a quotient of linearly independent solutions of u′′ + pu = 0, its valence sup c∈C∪{∞} # { z ∈ D : f(z) = c}, equals the oscillation number sup solutions u6≡0 # { z ∈ D : u(z) = 0}, of that equation. In particular, both quantities are finite or both infinite. The equation u′′ + pu = 0 in D has finite oscillation number if p is bounded. Indeed, in view of Sturm’s theorem below and the standard method summarized in (i) of Theorem 10 (see Section 1), a bound |p| ≤ C implies that any two zeroes of a nontrivial solution are at least π/ √ C units apart. Boundedness, however, is not a necessary condition. Using a method of Nehari [11], Schwarz [14] has shown 2000 Mathematics Subject Classification: Primary 34M10, 34C10, 30C55.