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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)
##### Keywords:
distribution of zeros; polynomials
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##### References:
 [1] DOI: 10.1016/S0022-247X(86)80012-9 · Zbl 0596.30012 · doi:10.1016/S0022-247X(86)80012-9 [2] DOI: 10.2307/2044842 · Zbl 0552.30002 · doi:10.2307/2044842 [3] Bellman, Differential-difference equations (1963) [4] Beckenbach, Inequalities (1961) · doi:10.1007/978-3-642-64971-4 [5] Yates, A Handbook on Curves and their Properties (1959) [6] DOI: 10.1093/qmath/36.2.255 · Zbl 0596.30011 · doi:10.1093/qmath/36.2.255 [7] DOI: 10.2307/2045450 · Zbl 0537.30005 · doi:10.2307/2045450 [8] Stolarsky, Coll. Math. Soc. Jdnos Bolyai 34 (1984) [9] Rahman, Canad. J. Math. 32 pp 1– (1980) · Zbl 0433.30005 · doi:10.4153/CJM-1980-001-0 [10] Rahman, Canad. J. Math. 30 pp 332– (1978) · Zbl 0381.30005 · doi:10.4153/CJM-1978-030-2 [11] Pólya, Problems and Theorems in Analysis 1 (1970) [12] Nulton, C. R. Math. Rep. Acad. Sci. Canada 6 pp 243– (1984) [13] Marden, Mathematical Surveys 3 (1966) [14] Nicolas, Cinquante Ans de Polynomes (1990) [15] Laguerre, Oeuvres 1 (1972) [16] Hardy, Inequalities (1964) [17] DOI: 10.1002/zamm.19820621007 · doi:10.1002/zamm.19820621007
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