On applications of the Schwarzian derivative in the real domain.

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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, Zbl0035.05104MR29999
3. 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, Zbl0066.32602MR69874
4. 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); 
5. 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. Zbl0065.07303
6. KLEIN, F., Gesammelte mathematische Abhandlungen, vol. 2, pp. 551-567, or [5], pp. 211-249. 
7. BIEBERBACH, L., Einführung in die Theorie der Differentialgleichungen im reellen Gebiet, 1956, pp. 228-233. Zbl0075.07101MR86188
8. KLEIN, F., Vorlesungen über die hypergeometrische Funkition, ed. 1933 Zbl0007.12202MR668700JFM59.0375.11
9. WINTNER, A., A priori Laplace transformations of linear differential equations, «American Journal of Mathématics», vol. 71 (1949), pp. 587-594. Zbl0040.34102MR30673
10. WINTNER, A., On the non-existence of conjugate points, ibid., vol. 73 (1951), pp. 368-380. Zbl0043.08703MR42005