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Suppose that

R(z)= \sum_{n=-\infty}^{\infty} a_nz^n

converges in a neighborhood of the unit circle

\{ z : \;|z|=1\ \}

in the complex plane, and

R(z)=P(z) / Q(z)

is a rational function in this neighborhood, where

P

and

Q

are polynomials of degree at most

k

. Prove that there is a constant

c

independent of

k

such that

\sum_{n=-\infty} ^{\infty} |a_n| \leq ck^2 \max_{|z|=1} |R(z)|.

H. S. Shapiro, G. Somorjai

Problem Statement