On the coefficients of rational functions

Abstract

Let f(z) be a rational function of order ≦n; i. e. the quotient of two polynomials of order ≦n. Suppose that it has no poles on the unit circle |z| = 1 and $\mathop {\max }\limits_{|z| = 1} |f(z)| = 1$. Let

$$f(z) = \sum\limits_{k = - \alpha }^\infty {a_k z^k }$$

be its Laurent expansion valid on the unit circle. G. Ehrung raised the problem of estimating $\sum\limits_{k = - \infty }^\infty {|a_k |}$ in terms of n and conjectured the upper bound const. n which, if true, would be best possible. H. S. Shapiro showed this (unpublished) if all the poles are inside the unit disc and in the general case for n replaced by n ² (oral communication). Here we prove by a different method the Theorem.

$$\sum\limits_{k = - \alpha }^\infty {|a_k |} \leqq 43n\sqrt {\log 3n} .$$

Proof. We have

$$a_k = (1/2\pi i)\int\limits_\Gamma {f(z)z^{ - k - 1} dz}$$

implying, restricting ourselves first to negative indices,

$$f(z) = \sum\limits_{k = - \alpha }^\infty {a_k z^k }$$

(1)

where Γ is any cycle (union of closed curves) lying inside the unit disk which surrounds every pole in |z| < 1 of f(z) exactly once.