Treffer: Sharp Inequalities for the Zeros of Polynomials and Power Series: Sharp inequalities for the zeros of polynomials and power series.

In this interesting paper under review, the author establishes a number of sharp estimates for the zeros of polynomials and power series. A sample result (Theorem 1) is the following. Let \(f(z):=z^n+\sum_{\nu=0}^{n-1}a_{\nu}z^{\nu}\) be a monic polynomial with zeros \(z_1,\cdots,z_n\). Then for \(k=1,\cdots,n\), \[ \sum_{\nu=1}^k| z_{\nu}| \leq k-1+M_0(f)\quad\text{and}\quad \sum_{\nu=1}^n| z_{\nu}| \leq n-2+ | | f| | _{\infty}, \] where \(M_0(f)\) denotes the Mahler measure of \(f\). Moreover, the author provides examples of classes of polynomials for which equality is attained in the above inequalities.