Abstract
Oscillation of solutions of f^{(k)} + a_{k-2} f^{(k-2)} + cdots + a_1 f' +a_0 f = 0 is studied in domains conformally equivalent to the unit disc. The results are applied, for example, to Stolz angles, horodiscs, sectors, and strips. The method relies on a new conformal transformation of higher order linear differential equations. Information on the existence of zero-free solution bases is also obtained.
Highlights
Introduction and ResultsThe classical univalence criterion due to Nehari [12] states that a locally univalent meromorphic function f in the unit disc D is one-to-one if its Schwarzian derivative S f = ( f / f ) − (1/2)( f / f )[2] satisfies |S f (z)|(1 − |z|2)2 ≤ 2 for all z ∈ D. 71 Page 2 of 19Nehari’s proof is based on the representation a = S( f1/ f2)/2 of the analytic coefficient of f +af = 0 (1)in terms of the quotient of its two linearly independent solutions f1 and f2
By restating [11, Proposition 1.4.7] with the aid of some basic properties satisfied by Wronskian determinants [11, Chap. 1.4], we see that the functions 1, y1, . . . , yk−1 are linearly independent meromorphic solutions of the differential equation y(k)
A is analytic in the right half-plane, and f + a f = 0 has linearly independent zero-free solutions f j (z)
Summary
The classical univalence criterion due to Nehari [12] states that a locally univalent meromorphic function f in the unit disc D is one-to-one if its Schwarzian derivative S f = ( f / f ) − (1/2)( f / f )[2] satisfies |S f (z)|(1 − |z|2)2 ≤ 2 for all z ∈ D
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