Vol. 19, No. 3, 1966

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On the zeros of a linear combination of polynomials

Robert Vermes

Vol. 19 (1966), No. 3, 553–559

In this paper we consider the location of the zeros of a complex polynomial f(z) expressed as f(z) = k=0nakpk(z) where {pk(z)} is a given sequence of polynomials of degree k whose zeros lie in a prescribed region E. The principal theorem states that the zeros of f(z) are in the interior of a Jordan curve S = {z;|F(z)| = Max(1,R)} where F maps the complement of E onto |z| > 1 and R is the positive root of the equation k=0n1λk|ak|tk λn|an|tn = 0, with λk > 0 depending on E only. Several applications of this theorem are given. For example; if {pk(z)} is a sequence of orthogonal polynomials on a z b, then we give an ellipse containing all the zeros of k=0nakpk(z).

Mathematical Subject Classification
Primary: 30.11
Received: 3 June 1965
Published: 1 December 1966
Robert Vermes