Bounded nonvanishing functions and Bateman functions

Abstract

We consider the family B-tilde of bounded nonvanishing analytic functions f(z) = a_0 + a_1 z + a_2 z^2 + ... in the unit disk. The coefficient problem had been extensively investigated, and it is known that |a_n| <= 2/e for n=1,2,3, and 4. That this inequality may hold for n in N, is know as the Kry\.z conjecture. It turns out that for f in B-tilde with a_0 = e^-t, f(z) << e^{-t (1+z)/(1-z)} so that the superordinate functions e^{-t (1+z)/(1-z)} = sum F_k(t) z^k are of special interest. The corresponding coefficient function F_k(t) had been independently considered by Bateman [3] who had introduced them with the aid of the integral representation F_k(t) = (-1)^k 2/pi int_0^pi/2 cos(t tan theta - 2 k theta) d theta . We study the Bateman function and formulate properties that give insight in the coefficient problem in B-tilde.