New inequalities for polynomials. (English) Zbl 0567.30006

Let \({\mathcal P}_ n\) be the class of polynomials \(P(z)=\sum^{n}_{\nu =0}a_{\nu}z^{\nu}\) of degree at most n. S. Berstein showed that (*) \(\| P'\| \leq n\| P\|\) and M. Riesz showed that \(M_ P(R)\leq R^ n\| P\|,\) \(R>1\), where \[ \| P\| =\max_{| z| =1}| P(z)|,\quad M_ P(R)=\max_{| z| =R}| P(z)|. \] In both cases we get equality only for \(P(z)=const. z^ n\). Thus we can expect to get sharper inequalities when one of the \(a_{\nu}\), \(\nu <n\), is known to be different from zero. The authors present several results which display this possibility. For example they show that \[ \| P'\| +\epsilon_ n| P(0)| \leq n\| P\|, \] where \(\epsilon_ 1=1\), and \(\epsilon_ n=2n/(n+2)\), \(n\geq 2\), and that \[ M_ P(R)+(R^ n-R^{n-2})| P(0)| \leq R^ n\| P\|, \] for \(n\geq 2\), and \(R>1\). The coefficient of \(| P(0)|\) is best possible for each n in the first case and for each R in the second. They give similar, more complicated results involving the next coefficient \(a_ 1=P'(0)\), and they say that their method can be used to study the dependence of \(\| P'\|\) and \(M_ P(R)\) on any other coefficients \(a_{\nu}\), \(\nu =2,...,n-1\). The method is based on studying the subclass \({\mathcal B}^ 0_ n\) of \({\mathcal P}_ n\) which consists of those polynomials Q which satisfy \(Q(0)=1\) and \(\| Q*P\| \leq \| P\|\) for all P in \({\mathcal P}_ n\), where Q*P denotes the Hadamard product of Q and P. Using results of T. Sheil- Small, J. Reine Angew. Math. 258, 137-152 (1973; Zbl 0267.30012) and the third author, Convolutions in geometric function theory (1982; Zbl 0499.30001) which connect \({\mathcal B}^ 0_ n\) with the class \({\mathcal R}\) of analytic functions f in \(| z| <1\) for which \(f(0)=0\) and Re f(z)\(>\), they show that a polynomial belongs to \({\mathcal B}^ 0_ n\) if and only if a certain matrix formed from its coefficients is positive semidefinite. The proofs of most of their results then rest on showing that appropriate matrices are positive semidefinite, a long but usually elementary task. Since polynomials in \({\mathcal B}^ 0_ n\) thus give rise to interesting inequalities the authors go on to give ways to manufacture such polynomials. As a by-product, they give an interpolation formula which shows that, in (*), \(\| P\|\) may be replaced by the maximum of P(z) at the (2n)th roots of unity. This well-written paper closes with several additional results, remarks and corollaries.
Reviewer: M.E.Muldoon


30C10 Polynomials and rational functions of one complex variable
30A10 Inequalities in the complex plane
26D05 Inequalities for trigonometric functions and polynomials
26D10 Inequalities involving derivatives and differential and integral operators
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