Referenze Bibliografiche
[1] In the classical writings, this connection is (sometimes tacitly) combined with what eventually became DARBOUX'S criterion (involving the image of the boundary of the Gomain J) for a schlicht mapping. The above-mentioned formulation of the classical fact (recently rediscovered, and used so as to supply sufficient criteria for schlicht behavior in general, by
NEHARI
[2], p. 545 and pp. 49-50), when applied to the particular case of schlicht triangle functions, was generalized by FELIX KLEIN to «oscillation theorems», which deal with a self-overlapping triangle and, correspondingly, replace a recourse to DARBOUX'S criterion by what corresponds to it in case of an arbitrairy Windungssahl ; cf. [3].
and
Z.
NEHARI
Univalent functions and linear differential equations, «
Lectures on Functions of a Complex Variable», Ann. Arbor,
1955, pp. 49-60 ; cf. also pp. 214-215 and Lemma 2 and Lemma 3 (and the earlier results of G. M. GOLUSIN and M. SCHIFFER, referred to in connection with those lemrnas) in a paper of A. RÉNYI, |
MR 69874 |
Zbl 0066.32602
Z.
NEHARI
On the geometry of conformal mapping, «Acta Scientiearum Mathematicarum» (Szeged), vol. 12 (1950), pp. 214-222. As I observed some time ago, NEHARI'S results become quite understandable (and, correspondingly, the proofs can be reduced considerably);
cf.
P.
HARTMAN
and
A.
WINTNER
,
On linear, second order differential equations in the unit circle, «
Transactions of the American Mathematical Society», vol.
78 (
1955), 493-495), if it is noticed that what is involved is precisely the distortion factor of the non-euclidean line element ds. |
Zbl 0065.07303[3]
F.
KLEIN
, Gesammelte mathematische Abhandlungen, vol. 2, pp. 551-567, or [5], pp. 211-249.
[4]
L.
BIEBERBACH
,
Einführung in die Theorie der Differentialgleichungen im reellen Gebiet,
1956, pp. 228-233. |
MR 86188 |
Zbl 0075.07101[6]
A.
WINTNER
,
A priori Laplace transformations of linear differential equations, «
American Journal of Mathématics», vol.
71 (
1949), pp. 587-594. |
MR 30673 |
Zbl 0040.34102[7]
A.
WINTNER
,
On the non-existence of conjugate points, ibid., vol.
73 (
1951), pp. 368-380. |
MR 42005 |
Zbl 0043.08703