A Theorem on Schlicht Functions

S. B. Townes
1954 Proceedings of the American Mathematical Society
In 1946 Friedman  found all schlicht functions, with certain restrictions, that have rational integral coefficients. It is the purpose of this article to find all schlicht functions similarly restricted, whose wth powers, n a positive integer, have coefficients belonging to an integral domain /, of characteristic zero, and that has no integer except zero of absolute value less than 1. 1. Notations and definitions. A function is said to be schlicht in a domain D if for any two points Zi and
more » ... two points Zi and z2, belonging to D, we have /(zi) =/(z2) only if 2i = 22. We shall seek all functions 00 /(z) = E a»z< -z + a222 + a&* + ---1 which are regular and schlicht within the circle | z\ < 1 and such that g(z)=/n(z) = Er0,z"+i = zn-r-01zn+1+02zn+2+ • • • have coefficients belonging to /. Let A(z) =zn/giz) have the expansion E dZl = 1 + CiZ + C222 + • • • . 0 2. Preliminary lemmas. Lemma 1. If g(z) has coefficients belonging to I, then A(z) is a polynomial with coefficients belonging to I, and conversely. The relationship between the coefficients of A(z) and those of g(z) is bi + ci = 0, , . h + bici + c2 = 0, (z. 1) bm + 0m_lCl + 0m_2C2 + • • • + Cm = 0. If the bi are integers of /, the d are also, and conversely. Let [/(z)/zh<"2 = 1 + E A<z*. I The Prawitz inequality , as revised by Bernardi [l], is