On Power Series Which are Bounded in the Interior of the Unit Circle II

The investigations communicated in the following paper are closely related to the theory of power series with positive real part, convergent in the interior of the unit circle, which was developed by C. Caratheodory [1] and supplemented in an important respect by O. Toeplitz [2]. On the basis of this theory, Caratheodory and Fejer [3] have already derived an interesting theorem about functions which are regular and bounded in the circle |x| < 1. In the following discussion, the theory of these functions is extended somewhat in several directions. This is not done with the aid of the Caratheodory results but in a direct way. The continued fraction algorithm introduced here very easily supplies an intrinsically important parametric representation for the coefficients of the power series to be considered. The principal content of the theory to be developed is already essentially present in Theorems 2 and 3 which concern this parametric representation and which have been proven in Section 3. Only a purely computational transformation of the expressions obtained is required in order to get from Theorem 2 to the main result of this paper, which is Theorem 8 in Section 6. From this theorem it is possible to deduce directly the Caratheodory-Toeplitz results; conversely the theorem also follows without difficulty from these results (cf. Section 8). The interesting Theorem 10 of Section 7, which appears here as a special case of Theorem 8, can also be proven easily with the aid of one of the important theorems of O. Toeplitz [4] about so-called “L-forms”, if one omits characterization of the limiting cases (Theorem 10, [4]).