A Theorem on Schlicht Functions

S. B. Townes
1954 Proceedings of the American Mathematical Society  
In 1946 Friedman [2] 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 [3], as revised by Bernardi [l], is
doi:10.2307/2032039 fatcat:saatpsfskjcgdnxmefvjecyufi