Bounded nonvanishing functions and Bateman functions

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

Functions in Bloch-type spaces and their moduli

Given a suitably regular nonnegative function $\omega$ on $(0,1]$, let $\mathcal B_\omega$ denote the space of all holomorphic functions $f$ on the unit ball $\mathbb B_n$ of $\mathbb C^n$ that satisfy $$|\nabla f(z)|\le C\frac{\omega(1-|z|)}{1-|z|},\qquad z\in\mathbb B_n,$$ with some fixed $C=C_f>0$. We obtain a new characterization of $\mathcal B_\omega$ functions in terms of their moduli.Comment: 9 pages; to appear in Ann. Acad. Sci. Fenn. Math. 41 (2016), No.

Domain of validity of Szegö quadrature formulas

AbstractAs is well known, the n-point Szegö quadrature formula integrates correctly any Laurent polynomial in the subspace span{1/zn-1,…,1/z,1,z,…,zn-1}. In this paper we enlarge this subspace. We prove that a set of 2n linearly independent Laurent polynomials are integrated correctly. The obtained result is used for the construction of Szegö quadrature formulas. Illustrative examples are given