1. Math. Ann. Vol. 64, 1907, p.95, and Rend. Di Palmermo, Vol. 32, 1911, p.193. The Caratheodory theorems have been proven in a different way by E. Fischer (Rend. Di Palermo, vol. 32, p.240), G. Herglotz (Leipz. Berichte, 1911, p.501), G. Frobenius (Berl. Berichte, 1912, p.16) and by the author (loc. cit., p.4). Cf. also F. Riesz, Ann. De L’Ecole Norm., 1911, p.33, and Journ. F. Math., Vol. 146, 1915, p.83.
2. Goett. Nachrichten, 1910, p.489, and Rend, di Palermo, Vol. 32, p. 191.
3. Rend, di Palermo, Vol. 32, p. 131–235. Cf. also T. H. Gronwall, Annals of Math., Vol. 16, 1914, p. 77, and G. Pick, Math. Ann., Vol. 77, 1915, p. 7.
4. Math. Ann., Vol. 70, 1911, p.351 (referring to Theorem V of this paper).
5. Nothing is assumed regarding the behavior of f(x) for |x| > 1.