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The zeros of a certain family of trinomials. (English) Zbl 0751.30007
The distribution of zeros of the polynomials \(T(z):=mz^ n-nz^ m+n-m\) and \(p_ n(z):=\lambda z^ n+1-\lambda-(\lambda z+1-\lambda)^ n\), where \(n>m>0\) are integers and \(\lambda\) is real, \(0<\lambda<1\), is studied. Let \(U\) be the open unit disc. First the authors prove that for large \(n\) the zeros of \(T(z)\) asymptotically are evenly distributed near \(\partial U\). The main result is: For \(n\leq 1+5/51\lambda\) \(p_ n(z)\) has no zero inside \(U\). If \(1/\lambda\) is not an integer, then for sufficiently large \(np_ n(z)\) has a zero inside \(U\). If \(1/\lambda\) is an integer \(>2\), then apart from a double zero at \(z=1\) all zeros of \(p_ n(z)\), for \(n\geq 475\), are outside \(\overline U\).
Reviewer: H.-J.Runckel (Ulm)

MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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