# The Krzyz Problem and Polynomials with Zeros on the Unit Circle Academic Article

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### abstract

• Let $\Pi_{n,M}$ be the class of all polynomials $p(z) = \sum _{0}^{n} a_{k}z^{k}$ of degree n which have all their zeros on the unit circle $|z| = 1$ , and satisfy $M = \max _{|z| = 1}|\,p(z)|$ . Let $\mu _{k,n} = \sup _{p\in \Pi _{n,M}} |a_{k}|$ . Saff and Sheil-Small asked for the value of $\overline {\lim }_{n\rightarrow \infty }\mu _{k,n}$ . We find an equivalence between this problem and the Krzyz problem on the coefficients of bounded non-vanishing functions. As a result we compute $$\overline {\lim }_{n\rightarrow \infty }\mu _{k,n} = {{M} \over {e}}\quad {\rm for}\ k = 1,2,3,4,5.$$ We also obtain some bounds for polynomials with zeros on the unit circle. These are related to a problem of Hayman.

### publication date

• January 1, 2002