On applications of the Schwarzian derivative in the real domain.
Wintner, Aurel
Bollettino dell'Unione Matematica Italiana, Tome 12 (1957), p. 394-400 / Harvested from Biblioteca Digitale Italiana di Matematica
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Publié le : 1957-09-01
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     author = {Aurel Wintner},
     title = {On applications of the Schwarzian derivative in the real domain.},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {12},
     year = {1957},
     pages = {394-400},
     zbl = {0080.06803},
     mrnumber = {95318},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_1957_3_12_3_394_0}
}

Wintner, Aurel. On applications of the Schwarzian derivative in the real domain.. Bollettino dell'Unione Matematica Italiana, Tome 12 (1957) pp. 394-400. http://gdmltest.u-ga.fr/item/BUMI_1957_3_12_3_394_0/

[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].

[2] Nehari, Z., The Schwarzian derivative and schlicht functions, «Bulletin of the American Mathematical Society», vol. 55 (1949), pp. 545-551, | MR 29999 | Zbl 0035.05104

and Nehari, Z.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

Nehari, Z.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.Hartman, P. and Wintner, A., 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] Klein, F., Gesammelte mathematische Abhandlungen, vol. 2, pp. 551-567, or [5], pp. 211-249.

[4] Bieberbach, L., Einführung in die Theorie der Differentialgleichungen im reellen Gebiet, 1956, pp. 228-233. | MR 86188 | Zbl 0075.07101

[5] Klein, F., Vorlesungen über die hypergeometrische Funkition, ed. 1933 | JFM 59.0375.11 | MR 668700 | Zbl 0007.12202

[6] Wintner, A., A priori Laplace transformations of linear differential equations, «American Journal of Mathématics», vol. 71 (1949), pp. 587-594. | MR 30673 | Zbl 0040.34102

[7] Wintner, A., On the non-existence of conjugate points, ibid., vol. 73 (1951), pp. 368-380. | MR 42005 | Zbl 0043.08703